SAP-1
1) If A= 2 -1
-1 2 and I is the unit matrix of order 2, then A² is equal to
a) 4A - 3I b) 3A - 4I c) A - I d) A + I
2) The multiplicative inverse of
2 1
7 4 is
a) 4 -1 b) 4 -1 c) 4 -7 d) -4 -1
-7 -2 -7 2 7 2 7 -2
3) For a real number α, let A(α) denote the matrix
cosα sinα
-sinα cosα
Then for real numbers α₁ and α₂, the value of A(α₁) A(α₂) is
a) A(α₁α₂) b) A(α₁ + α₂) c) A(α₁ - α₂) d) A(α₂ -α₁)
4) If the system of equations x+ 2y + 3z= 1, 2x + ky + 5z= 1, 3x+ 4y + 7z= 1 has no solutions, then
a) k= -1 b) k= 1 c) k= 3 d) k= 2.
5) Assuming that the sums and products given below are defined, which of the following is not true for matrices?
a) AB= AC does not imply B= C
b) A+ B= B+ A
c) (AB)'= B'A'
d) AB= O implies A= O or B= O.
6) If A=1. 0 2 & Adj A= 5 a -2
-1 1 -2 1 1 0
0 2 1 -2 -2 b
Then the values of a and b are
a) a= -4, b= 1 b) a= -4, b= -1 c) a= 4, b= 1 d) a= 4, b= -1
7) If A= -1 0
0 2 then the value of A³- A² is
a) I b) A c) 2A d) 2I.
8) If 1, w, w² are the cube roots of unity then the value of m for which the matrix
1. w m
w m 1
m 1 w is singular, is
a) 1 b) -1 c) w d) w².
9) If A= -x - y
z t, then transpose of adj A
a.) t z b) t y c) t -z d) none
-y -x -z -x y -x.
10) If A square metrix of order 3x3 and λ is a scalar, then adj(λA) is equal to
a) λ adj A B) λ² adj A c) λ⅗ adj A d) λ³ adj A.
11) The inverse of 5 -2
3 1
a) -2/13 5/13 b) 1 2 c) 1/11 2/11 d) 1 3
1/13 3/13 -3 5 -3/11 5/11 -2 5
12) If A=3 5 & B= 1 17
2 0. 0 -10
a) 80 b) 100 c) -110 d) 92.
CONICS
1. Conic sections:
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.
a) The fixed point is called the focus.
b) The fixed straight line is called the directrix.
c) The constant ratio is called the eccentricity denoted by e.
d) The line passing through the focus & perpendicular to the directrix is called the axis.
e) A point of intersection of a conic with its axis is called vertex.
2. General equation of a conic: Focal Directrix Property:
The general equation of a conic with focus (p,q) & directrix lx + my + n=0 is
(l²+ m²)[(x - p)²+ (y - q)²]= e²(lx + my + n)² ≡ ax¹+ 2hxy+ by²+ 2gx + 2fy + c=0
3. Distinguish between the Conic:
The nature of the conic section depends upon the position of the focus S w.r.t. the directrix and also upon the value of the eccentricity e. Two different cases arise.
Case(i): when the focus lie on the directrix:
In this case D≡ abc+ 2fgh - af²- bg² - ch²=0 and the general equation of a conic represent a pair of straight lines and if:
e> 1 the line will be real and distinct intersecting at S.
e= 1 The lines will be coincident.
e < 1 The lines will be imaginary.
Case(ii): When the focus does not lie on the directrix:
e=1; D≠ 0 and h²= ab a parabola
0< e < 1; D≠ 0 and h²< ab an ellipse
D≠ 0; e> 1 and h²> ab. a hyperbola
e> 1; D≠ 0 and h²> ab; a+ b = 0 . A rectangular hyperbola
4. PARABOLA
A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix).
Standard equation of a parabola is y²= 4ax. For this parabola:
i) Vertex is (0,0)
ii) Focus is (a,0)
iii) Axis is y=0
iv) Directrix is x+ a=0
a) Focal distance:
The distance of a point on the parabola from the focus is called the focal distance of the point.
b) Focal chord:
A chord of the parabola, which passes through the focus is called a focal chord.
c) Double ordinate:
A chord of the parabola perpendicular to the axis of the symmetry is called a double ordinate.
d) Latus rectum:
A double ordinate passing through the focus of a focal chord perpendicular to the axis of parabola is called the latus rectum. For y²= 4ax.
• Length of the latus rectum= 4a.
• Length of the semi latus rectum= 2a.
• Ends of the latus rectum= L(a, 2a) & L'(a, -2a)
Notes that:
i) Perpendicular distance from focus on directrix= half the latus rectum.
ii) Vertex is middle point of the focus and the point of intersection of directrix and axis.
iii) Two parabola are said to be equal if they have the same latus rectum.
5. Parametric Representation:
The simplest and the best form of representing the coordinates of a point on the parabola is (at², 2at). The equation x= at² & y= 2at together represents the parabola y²= 4ax, t being the parameter.
6. Type of PARABOLA
Four standards forms of the parabola are y²= 4ax; y²=- 4ax; x²= 4ay; x²= -4ay
When parabola y²= 4ax
Vertex: (0,0)
Focus: (a,0)
Axis: y= 0
Directrix: x= - a
Length of latus rectum: 4a
Ends of Latus rectum: +a, ±2a)
Parametric equation: (at²,2at)
Focal length: x + a
When parabola y²= - 4ax
Vertex: (0,0)
Focus: (- a,0)
Axis: y= 0
Directrix: x= a
Length of latus rectum: 4a
Ends of Latus rectum: (-a, ±2a)
Parametric equation: (-at²,2at)
Focal length: x - a
When parabola x²= 4ay
Vertex: (0,0)
Focus: (0,a)
Axis: x = 0
Directrix: y= - a
Length of latus rectum: 4a
Ends of Latus rectum: (±2a, a)
Parametric equation: (2at², at²)
Focal length: y + a
When parabola x²= - 4ay
Vertex: (0,0)
Focus: (0, - a)
Axis: x = 0
Directrix: y= a
Length of latus rectum: 4a
Ends of Latus rectum: (±2a, - a)
Parametric equation: (2at², - at²)
Focal length: y - a
When parabola (y - k)²= 4a(x - h)
Vertex: (h+ a, k)
Focus: (0,a)
Axis: y = k
Directrix: x+ a - h= 0
Length of latus rectum: 4a
Ends of Latus rectum: (h+ a, k± 2a)
Parametric equation: (h+ at², k+ 2at)
Focal length: x - h + a
When parabola (x - o)²= 4b(y - q)
Vertex: (p,q)
Focus: (p, b+ q)
Axis: x = p
Directrix: y+ b - q = 0
Length of latus rectum: 4apb
Ends of Latus rectum: (p± 2a, q+ a)
Parametric equation: (p+ 2at, q+ at²)
Focal length: y - q + b
7. Position of a point relative to a parabola:
The point (x₁, y₁) lies outside, on or inside the parabola y²= 4ax according as the expression y₁²- 4ax₁ is positive, zero or negative.
8. Chord joining two points:
The equation of a chord of the parabola y²= 4ax joining its two points P(t₁) and Q(t₂) is
y(t₁ + t₂) = 2x + 2at₁t₂
Note:
i) If PQ is focal chord then t₁t₂= -1.
ii) Extremities of focal chord can be taken as (at², 2at) and (a/t², -2a/t).
9. LINE AND A PARABOLA :
a) The line y= mx + c meets the parabola y²= 4ax in two points real, coincident or imaginary according as > = < cm => condition of tangency is, c= a/m.
NOTE: Line y= mx + c will be tangent to parabrx²= 4ay if c= - am².
b) Length of the chord intercepted by the parabola y²= 4ax on the line y= mx + c is: (4/m²) √=a(1+ m¹)(a - mc)}.
NOTE
Length of the focal chord making an angle α with x-axis is 4a coesec²α.
10. LENGTH OF SUBTANGENT & SUBNORMAL
PT and PG are the tangent and normal respectively at the qP to the parabola y½= 4ax. Then
TN= length of subtangent= twice the abscissa of the point P (Subtangent is always bisected by the vertex)
NG= length of subnormal which is constant for all points on the parabola and equal to its semilatus rectum (2a).
11. TANGENT TO THE PARABOLA y²= 4ax:
a) Point form:
Equation of tangent to the given parabola at its point (x₁ y₁) is
yy₁ = 2a(x + x₁)
b) Slope form:
Equation of tangent to the given parabola whose slope is 'm', is
y= mx + a/m, (m≠ 0)
Point of contact us (a/m², 2a/m)
c) Parametric form:
Equation of tangent to the given parabola at its point P(t), is ty= x + at²
NOTE:
Point of intersection of the tangents at the point t₁ & t₂ is [at₁t₂, a(t₁ + t₂)]
12. NORMAL TO THE PARABOLA y²= 4ax:
a) Point form:
Equation of normal to the given parabola at its point (x₁, y₁) is
y- y₁ = - y₁(x - x₁)/2a.
b) Slope form:
Equation of normal to the given parabola whose slope is 'm', is
y= mx - 2am - am³
foot of the normal is (am², - 2am)
c) Parametric form:
Equation of normal to the given parabola at its point P(t), is
y+ tx = 2at + at²
NOTE
i) Point of intersection of normals at t₁ & t₂ is [a(t₁²+ t₂²+ t₁t₂+2), - at₁t₂ (t₁ + t₂))
ii) If the normal to the parabola y²= 4ax at the point t₁, meets the parabola again at the point t₂ then t₂ = - (t₁ + 2/t₁).
iii) If normals to the parabola y²= 4ax at the point t₁ & t₂ intersect again on the parabola at the point 't₃' then t₁ t₂ = 2; t₃ = -(r₁ + t₂) and the line joining t₁ and t₂ passes through a fixed point (-2a, 0).
i.e., am³+ m(2a - h)+ k =0
This gives m₁ + m₂+ m₃= 0; m₁ m₂+ m₂m₃+ m₃m₁= (2a - h)/a; m₁m₂m₃= - k/a
Where m₁ , m₂, m₃ are the slopes of the three concurrent normas:
• Algebraic sum of slopes of the three concurrent normals is zero.
• Algebraic sum of ordinates of the three co-normal points on the parabola is zero.
• Centroid of the ∆ formed by three consecutive normal points lies on the axis of parabola (x-axis).
13. AN IMPORTANT CONCEPT:
If a family of straight lines can be represented by an equation λ²P+ λQ+ R= 0 where λ is a parameter and P,Q,R are linear functions of x and y then the family of lines will be tangent to the curve Q²= 4PR.
14. PAIR OF TANGENTS:
The equation of the pair of tangents which can be drawn from any point P(x₁ , x₂) outside the parabola to the parabola y²= 4ax is given by: SS₁ = T² where
S= y²- 4ax; S₁= y₁² - 4ax₁; T ≡ yy₁ - 2a(x + x₁).
15. DIRECTOR CIRCLE:
Locus of the point of intersection of the perpendicular tangents to the parabola y²= 4ax is called the director circle. It's equation is x+ a = 0 which is parabola's own directrix.
16. CHORD OF CONTACT:
Equation of the chord of contact of tangents drawn from a point P(x₁, y₁) is yy₁ = 2a(x + x₁)
NOTE
The area of the triangle formed by the tangents from the point (x₁, y₁) and the chord of contact us (y₁²- 4ax₁)³⁾²/2a i.e., (S₁)³⁾²/2a, also note that the chord of contact exists only if the point P is not inside.
17. CHORD WITH A GIVEN MIDDLE POINT
Equation of the chord of the parabola y²= 4ax whose middle point is x₁, y₁ is
y - y₁= 2a(x - x₁)/y₁.
This reduced to T= S₁ where T≡ 2a(x + x₁) and S₁ = y₁²- 4ax₁.
18 DIAMETER:
The locus of the middle point of a system of parallel chords of a parabola is called a DIAMETER. Equation to the diameter of a parabola is y= 2a/m, where m= slope of parallel chords.
19. IMPORTANT HIGHLIGHTS:
a) If the tangent and normal at any point P of the parabola intersect the axis at T and G then ST= SG= SP where S is the locus. In other words the tangent and the normal at a point P on the parabola are the bisectors of the angle between the focal radius SP and the perpendicular from P on the directrix. From this we conclude that all rays emanating from S will become parallel to the axis of the parabola after reflection.
b) The portion of a tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.
c) The tangents at the extremities of a focal chord intersect at right angles on the directrix, and a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P(at², 2at) as diameter touches the tangents at the vertex and intercepts a chord of length a√(1+ t²) on a normal at the point P.
d) Any tangent to a parabola and the perpendicular on it from the focus meet on the tangent at the vertex.
e) Semi latus rectum of the parabola y²= 4ax, is the harmonic mean between segments of any focal chord of the parabola is; 2a = 2bc/(b + c) i.e., 1/b + 1/c = 1/a.
f) If the tangent at P and Q meet in T, then:
i) TP and TQ subtend equal angles at the focus S.
ii) ST²= SP. SQ
iii) The triangles SPT and STQ are similar.
g) Tangents and Normals at the Extremities of the latus rectum of a parabola y²= 4ax constitue a square, their points of intersection being (-a,0) and (3a,0).
NOTE
i) The two tangents at the Extremities of focal chord meet on the foot of the directrix.
ii) Figure LNL'G is a square of side 2√2 a.
h) The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus.
Exercise -1
1) Find the locus of a point which moves such that its distance from the point (0,1) is twice its distance from the line 3x + 4y +1=0.
2) What conic does the equation 25(x²+ y²- 2x +1)= (4x -3y +1)² represent ?
Exercise -2
1) What conic does 13x²+ 37y²+ 2x +14y - 18xy -2=0 represent ?
2) What conic is represented by the equation √(ax)+ √(by)= 1?.
3) if the equation x² - y²- 2x +2y + K=0 represent a degenerate conic then find the value of K.
4) If the equation x²+ y²- 2x -2y + c =0 represents an empty set then find the value of c.
5) If the equation of conic 2x²+ 3y² -3x +5y + xy +K =0 represent a single point, then find the value of K.
6) For what value of K the equation of conic 4x -6y +2xy + K =0 represents two intersecting straight lines ? If K= 17 then , this equation represent ?
EXERCISE -3
1) Find the centre of the conic 14x²- 4xy + 11y²- 44x - 58y + 71=0.
Exercise - 4
Exercise - 5
1) Find the equation of the parabola whose focus is at (-1,-2) and directrix is the straight line x - 2y +3=0.
2) Find the equation of the parable whose focus is (4,-3) and vertex is (4,-1).
3) The focal distance of a point on a parabola y²= 8x is 8. find it.
4) QQ' is a double ordinate of a parabola y²= 4ax. Find the locus of the point of trisection .
5) Prove that area of the triangle inscribed in the parabola y²= 4ax is (1/8a)(y₁ - y₂)(y₂ - y₃)(y₃ - y₁), where y₁, y₂, y₃ are the ordinates of the vertices.
6) Find the length of the side of an equilateral triangle inscribed in the parabola y²= 4ax, so that one angular point is at the vertex.
7) Prove that the equation of the parabola whose focus is (0, 0) and tangent at the vertex is x - y+1=0 is x²+ y² -4x + 4y +2xy - 4=0.
8) Find the equation of the parabola whose latus rectum is 4 units , axis is the line 3x + 4y -4=0 and the tangent at the vertex is the line 4x - 3y +7=0.
9) Find the vertex, focus, latuce rectum , axis and the directrix of the parabola x²+ 8x +12y +4 =0.
10) Prove that the equation y²+ 2ay +2by + c=0 represents a parabola whose axis is parallel to the axis of x . Find its vertex.
11) The x and y coordinates of any point P are expressed as x= (V cosk) t, y= (V sink) t - (1/2) gt², where t is a parameter and V, k, g are constants. Show that the locus of the point P(x,y) is a parabola. Find the coordinates of the vertex of the parabola.
12) Find the equation of the parabola with its vertex at (3,2) and its focus at (5,2).
13) Find equation of the parabola with the latus rectum joining the point (3,6) and (3,-2).
14) Find the equation of the parabola whose axis parallel to the y-axis and which passes through the points (0,4),(1,9) and (4,5) and determine its latus rectum.
OBJECTIVE - 1
1) The vertex of the parabola y²+ 6x - 2y +13=0 is:
a) (-2,1) b) (2,-1) c) (1,1) d) (1,-1)
2) if the parabola y²= 4ax passes through (3,2) then the length of latuce rectum is:
a) 1/3 b) 2/3 c) 1 d) 4/3
3) The value of p such that vertex of y= x²+ 2px + 13 is 4 units above the x-axis is:
a) ±2 b) 4 c) ±3 d) 5
4) The length of the latus rectum of the parabola whose focus is (3,3) and directrix is 3x - 4y -2=0, is:
a) 1 b) 2 c) 4 d) 8
5) If the vertex and focus of a parabola are (3,3) and (-3,3) respectively, then its equation is:
a) x²- 6x + 24y -63=0
b) x²- 6x + 24y + 81=0
c) y²- 6y + 24x + 63=0
d) y²- 6y - 24x + 81=0
6) If the vertex of the parabola y= x²- 8x + c lies on x-axis, then the value of c is:
a) 4 b) -4 c) 16 d) -16
7) The parabola having its focus at (3,2) and directrix along the y-axis has its vertex at:
a) (3/2,1) b) (3/2,2) c) (3/2,1/2) d) (3/2,-1/2)
8) The directrix of the parabola x²- 4x - 8y +12 =0 is :
a) y=0 b) x=1 c) y=-1 d) x = -1
9) The equation of the latus rectum of the parabola x²+ 4x + 2y =0 is:
a) 3y -2=0 b) 3y + 2=0 c) 2y -3=0 d) 2y + 3 =0
10) trhe focus of the parabola x²- 8x + 2y +7 =0 is:
a) (0,,-1/2) b) (4,4) c) (4,9/2) d) (-4,-9/2)
11) The equation of the parabola with the focus (3,0) and directrix x +3=0 is:
a) y²= 2x b) y²= 3x c) y²= 6x d) y²= 12x
12) Equation of the parable whose axis is parallel to y-axis and which passes through the points (1,0), (0,0) and (-2,4) is:
a) 2x²+ 2x = 3y
b) 2x²- 2x = 3y
c) 2x²+ 2x = y
d) 2x²- 2x = y
Subjective - 1
1) Find the equation of the parabola whose focus is (3,5) and directors is to the line 3x - 4y +1=0.
2) Find the equation of the parabola is Focus is at (-6,-6) and vertex is at (-2,2).
3) Find the vertex, focus, axis , directrix and latus rectum of the parabola 4x²+ 12x - 20y + 67 = 0.
4) Find the name of the conic represented by √(x/a) + √(y/b)= 1.
5) Determine the name of the curve described parametrically by the equation x= t²+ t+1, y= t²- t +1.
6) Prove that equation of the parabola whose vertex and focus are on the x-axis at a distance a and A' from the origin respectively is y²= 4(A' - a)(x - a).
7) Find the equation is parabola whose axis is parallel to x-axis and which passes through the point (0,4),(1,9) and (-2,6). Also , find the letus rectum.
8) The equation ax²+ 4xy + y²+ ax + 3y +2 = 0 represents a parabola then find the value of a.
EXERCISE - 2
1) Show that the point (2,3) lies outside the parabola y²= 3x.
2) Find the position of the point (-2,2) with the respect to the parabola y²- 4y + 9x +13=0.
3) if the point (at², 2at) be the externalty of a focal chord of parabola y² 4ax then show that the length of the focal chord is a (t + 1/t)².
4) Prove that the semilatus rectum of the parabola y²= 4ax is the harmonic mean between the segments of any focal chord of the parabola.
5) Show that the focal chord of parabola y²= 4ax makes an angle α with the x-axis is of length 4a cosec²α.
6) Prove that the length of a focal chord of a parabola varies inversely as the square of its distance from the vertex.
7) Show that the straight line lx + my + n=0 touches the parabola y²= 4ax if ln = am².
8) Show that the line x cosα + y sinα = p touches the parabola y²= 4ax if p cosα + a sin²α =0 and than the point of contact is (a tan²α, -2a tanα).
9) Show that the line x/l + y/m = 1 touches the parabola y²= 4a(x + b) if m²(l + b) + al²= 0.
10) Find the equations of the straight lines touching both x²+ y²= 2a² and y²= 8ax.
11) Find the equation of the common tangents to the parabola y²= 4ax and x²= 4by.
12) The tangents to the parabola y²= 4ax make angle θ₁ and θ₂ with x-axis. Find the locus of their point of intersection if cotθ₁ + cot θ₂ = c.
13) Show that the locus of the points of intersection of the mutually perpendicular tangents to a parabola is the directrix of the parabola.
14) The tangents to the parabola y²= 4ax at P (at²₁, 2at₁) and Q(at²₂, 2at₂) intersect at R. Prove that the area of the triangle PQR is (1/2) a²|(t₁ - t₂)|³.
15) Show that normal to the parabola y²= 8x at the point (2, 4) meets it again at (18, 12). Find also the length of the normal chord.
16) Prove that the chord y- x √2 + 4a √2= 0 is a normal chord of the parabola y²= 4ax. Also , find the point on the parabola when the given chord is normal to the parabola.
17) If the normal to the parabola y²= 4ax, makes an angle with θ with the axis show that it will cut the curve again at an angle tan⁻¹((1/2) tanθ).
18) Prove that the normal chord to a parabola y²= 4ax at the point whose ordinate is equal to abscissa subtends a right angle at the focus.
19) Show that the locus of points such that two of the three normals drawn from them to the parabola y²= 4ax coincide is 27ay²= 4(x - 2a)³.
20) Find the locus of the point through which pass three normals to the parabola y² = 4ax such that two of them make angles θ₁ and θ₂ respectively with the axis such that tanθ₁ tan θ₂= 2.
21) If the three normal from a point to the parabola y²= 4ax cut the axis in points whose distance from the vertex are in AP, show that the point lies on the curve 27ay²= 2(x - 2a)³.
22) The normal at P, Q, R on the parabola y²= 4ax meet in the point on the line y= k. Prove that the sides of the triangle PQR touch the parabola x²- 2ky = 0.
23) Find the point on the axis of the parabola 3y²+ 4y - 6x +8=0 from when three distinct normals can be drawn.
24) A circle cuts the parabola y²= 4ax at right angles and passes through the focus, show that its centre lies on the curve y²(a + 2x)= a(a+ 3x)².
OBJECTIVE - 2
1) If2x + y + λ = 0 is a normal to the parabola y²= -8x then value of λ is:
a) -24 b) -16 c) - 8 d) 24
2) The slope of a chord of the parabola y²= 4ax which is normal at one end and which subtends a right angle at the origin is
a) 1/√2 b) √2 c) -1/√2 d) -√2
3) The common tangent to the parabola y²= 4ax and x²= 4ay is
a) x+ y + a=0
b) x+ y - a=0
c) x- y + a=0
d) x - y - a=0
4) The circle x²+ y²+ 4λx = 0 which λ ∈ R touches the parabola y²= 8x. The value of λ is given by:
a) λ∈ (0, ∞)
b) λ∈ (-∞, 0)
c) λ∈ (1, ∞)
d) λ∈ (-∞, 1)
5) If the normal at two points P and Q of a parabola y²= 4ax intersect at a third point R on the curve, then the product of ordinates of P and Q is
a) 4a² b) 2a² c) -4a² d) 8a²
6) The normals at three points P, Q, R of the parabola y²= 4ax meet in (h,k). The centroid of triangle PQR lies on
a) x= 0 b) y=0 c) x = - a d) y= a
7) The set of points on the axis of the parabola y²- 4x - 2y +5=0 from which all the three normals to the parabola are real is
a) (λ,0); λ > 1
b) (λ,1); λ > 3
c) (λ,2); λ > 6
d) (λ,3); λ > 8
SUBJECTIVE - 2
1) Show that any three tangents to a parabola whose slope are in harmonic progression enclose a triangle of constant area
2) A chord of parabola y²= 4ax subtends a right angle at the vertex. Find the locus of the point of intersection of tangents at its extremities.
3) Find the equation of the normal to the parabola y²= 4x which is:
a) parallel to the line y= 2x -5
b) Perpendicular to the line 2x + 6y +5=0.
4) The ordinates of a point P and Q on the parabola y²= 12yx are in the ratio 1:2. Find the locus of the point of intersection of the normal to the parabola at P and Q.
5) The normals at P, Q, R on the parabola y²= 4ax meet in a point on the line y= c. Show that the sides of the triangle PQR touch the parabola x²= 2cy.
6) The normals are drawn from (2λ,0) to the parabola y²= 4x. Show that λ must be greater than 1. One normal is always the x-axis. Find λ for which the other two normals are perpendicular to each other.
MARKS- 30 Time: 45 minutes 1x24= 24
1) If the determinant
P= 1 α 3
1 3 3
2 4 4 is adjoint of a 3x3 Matrix A and |A|= 4 then α is equals to
a) 11 b) 5 c) 0 d) 4
2) The term endependent of x in expansion of
{(x +1)/(x²⁾³ - x¹⁾³+1) - (x -1)/(x - x¹⁾²)}¹⁰ is
a) 120 b) 210 c) 310 d) 4
3) If the determinants
x 1 1 & B= x 1
A=1 x 1 1 x then dA/dx=
a) 3B+1 b) 3B c) -3B d) 1- 3B
4) The value of C₀ + 2C₁ + 3C₂ +....+ (n+1) Cₙ = 576, then n is
a) 7 b) 5 c) 6 d) 9
5) The remainder when, (10¹⁰+ 1)(10¹⁰+ 2) is divided by 6 is
a) 2 b) 4 c) 0 d) 6
6) If (1+ x + x²)ⁿ = 1+ a₁x + a₂x²+ ....+ a₂ₙx²ⁿ, then 2a₁ - 3a₂+ ... -(2n +1)a₂ₙ =
a) n b) -nl c) n +1 d) -nl -1
7) The value of x satisfying the equation of determinant
Cos2x sin2x sin2x
Sin2x cos2x sin2x= 0
Sin2x sin2x cos2x
And x ∈[0,π/4] is
a) π/4 b) π/2 c) π/16 d) π/8
8) If t₅, t₁₀, t₂₅ are 5ᵗʰ, 10ᵗʰ, and 25ᵗʰ terms of an AP respectively, then the value of determinant
t₅ t₁₀ t₂₅
5 10 25
1 1 1 is equal to
a) -40 b) 1 c) -1 d) 0
9) Five dice are tossed. What is the probability that five numbers shown will be different ?
a) 5/24 b) 5/18 c) 5/27 d) 5/81
10) if the events A and B are independent and if P(A')= 2/3, P(B)= 2/7 then P(A∩B) is equals to
a) 4/21 b) 3/21 c) 5/21 d) 1/21
11) Let P= [aᵢⱼ] be a 3 x 3 matrix and let Q= [bᵢⱼ], where bᵢⱼ = 2 ᶦ⁺ʲ aᵢⱼ for 1 ≤ i, j ≤ 3. If the determinant of P is 2, then the determinant of the matrix Q is
a) 2¹⁰ b) 2¹¹ c) 2¹² d) 2¹³
12) x(xⁿ⁻¹ - nαⁿ⁻¹) + αⁿ (n -1) is divisible by (x - α)² for
a) n> 1 b) n > 2 n ∈ N d) none
13) The sum of the series 1+ 3²/2! + 3⁴/4! + 3⁶/6!+....to ∞ is
a) e⁻³ b) e³ c) (1/2)(e³ - e⁻³) d) (1/2) (e³ + e⁻³)
14) Value of the series x/1.2 + x²/2.3 + x³/3.4 +.... is
a) 1- {(1- x)/x} log(1- x)
b) 1- {(1- x)/x} log(1+ x)
c) 1 + {(1- x)/x} log(1- x) d) none
15) Let the coefficient of powers of x in the 2ⁿᵈ, 3ʳᵈ and 4ᵗʰ terms in the expansion of (1+ x)ⁿ, where n is a positive integer, be in Arithmetic progression. The sum of the co-efficients of odd powers of x in the expansion is
a) 32 b) 64 c) 128 d) 256
16) The sum of the infinite series
1+ 1/3 + 1.3/3.6 + 1.3.5/3.6.9 + 1.3.5.7/3.6.9.12+ .....is equal to
a) √2 b) √3 c) √(3/2) d) √(1/√3))
17) The number of real values of K for which the system of equations
x+ 3y + 5z = Kx
5x+ y + 3z = Ky
3x+ 5y + z = Kz
has infinity number of solution is
a) 1 b) 2 c) 4 d) 6
18) Let Sₖ be the sum of an infinite GP series whose first term is K and common ratio is K/(K +1) (K> 0). Then the value of ∞ₖ₌₁∑ (-1)ᴷ/Sₖ is equal to
a) log4 b) log2 -1 c) 1- log2 d) 1- log4
19) Let A and B be two events with P(A')= 0.3, P(B)= 0.4 and P(A ∩B) = 0.5 then P(B/A U B') is equal to
a) 1/4 b) 1/3 c) 1/2 d) 2/3
20) Three numbers are choosen at random without replacement from {1, 2, 3, .....,8}. The probability that their minimum is 3, given that their maximum is 6, is
a) 1/4 b) 2/5 c) 3/8 d) 1/5
21) If C₀, C₁, C₂, C₃,.... are binomial coefficients in the expansion of (1+ x)ⁿ. Then C₀/3 - C₁/4 + C₂/5 - C₃/6+.... is equal to
a) 1/(n +1) - 2/(n+2) + 1/(n +3)
b) 1/(n +1) + 2/(n+2) - 1/(n +3)
c) 1/(n +1) - 1/(n+2) + 1/(n +3)
d) 2/(n +1) - 1/(n+2) + 2/(n +3)
22) If the matrix
A= a x
y a and xy=1. Then det(AA') is equal to
a) a²-1 b) (a²+1)² c) 1- a² d) (a²-1)²
23) Let A and B any two events. Which one of the following statements is always true ?
a) P(A'/B) = P(A/B)
b) P(A'/B) = P(B'/A)
c) P(A'/B) = 1- P(A/B)
d) P(A'/B) = 1- P(A/B')
24) The inverse of a symmetric matrix is
a) skew symmetric
b) symmetric
c) diagonal matrix d) none
Equation of Straight line
Sap-1
1) The number of points on x-axis which are at a distance c(c< 3) from the point (2,3) is
a) 2 b) 1 c) infinite d) no point
2) The distance between the points P(a cosα, a sinα) and Q(a cosβ, a sinβ) is
a) 4a sin{(α-β)/2} b) 2a sin{(α + β)/2} c) 2a sin{(α-β)/2} d) 2a cos{(α-β)/2}.
3) Determine the ratio in which y - x + 2 divides the line joining (3,-1) and (8,9).
4) If (1,4) is the centroid of a triangle and its two vertices are (4,-3) and (-9,7) then third vertices is
a) (7,8) b) (8,8) c) (8,7) d) (6,8).
5) The vertices of a triangle are A(0.-6), B(-6,0) and C(1,1), respectively, then coordinates of the excentre opposite to vertex A is.
a) (-3/2,-3/2) b) (-4,3/2) c) (-3/2,3/2) d) (-4,6).
6) If the vertices of a triangle are (1,2),(4,-6) and (3,5) then the area is
a) 25/2 b) 12 c) 5 d) 25.
7) The point A divides the join of the points (-5.1) and (3,5) in the ratio k: 1 and coordinates of points B and C are (1,5) and (7,-2) respectively. If the area of ∆ ABC be 2 units, then k equals to
a) (7,9) b) (6,7) c) 7,31/9 d) 9,31/9.
8) Show that the coordinates of the vertices of an equilateral triangle can not be rational.
9) The ends of the rod of length l moves on two mutually perpendicular lines, find the locus of the point on the rod which divides it in the ratio m₁: m₂
a) m₁²x²+ m₂²y²= l²/(m₁ + m₂)²
b) (m₂x)²+ (m₁y)²= {(m₁m₂l)/(m₁ + m₂)}²
c) (m₁x)²+ (m₂y)²= {(m₁m₂l)/(m₁ + m₂)}²
d) none.
10) If A(a,0) and B(-a,0) are two fixed points of ∆ ABC. If its vertex C moves in such way that cotA + cotB= λ, where λ is a constant, then the locus of the point C is
a) yλ = 2a b) y= λa c) ya = 2λ d) none
11) The equation of the lines which passes through the point (3,4) and the sum of its intercept on the axes is 14 is
a) 4x - 3y= 24, x - y= 7
b) 4x + 3y= 24, x + y= 7
c) 4x + 3y=- 24, x + y=- 7
d) 4x - 3y= -24, x - y=- 7.
12) Two points A and B move on the positive direction of x-axis and y-axis respectively, such that OA+ OB= K. Show that the locus of the foot of the perpendicular from the origin O on the line AB is (x + y)(x²+ y²)= Kxy.
13) Find the equation of the straight line on which the perpendicular from origin makes an angle 30° with x-axis and which forms a triangle of area (50/√3) square. units with the coordinates axes.
14) Equation of a line which passes through point A(2,3) and makes an angle of 45° with x-axis. If this line meet the line x+ y+1=0 at point P then distance AP is
a) 2√3 b) 3√2 c) 5√2 d) 2√5.
15) A variable line is drawn through O, to cut two fixed straight lines L₁ and L₂ in A₁ and A₂ respectively. A point A is taken on the variable line such that (m+ n)/OA = m/OA₁ + n/OA₂.
Show that the locus of A is a straight line passing through the point of intersection of L₁ and L₂ where O is being the origin.
16) A straight line through P(-2,-3) cuts the pair of straight line x²+ 3y²+4xy - 8x - 6y - 9= 0 in Q and R. Find the equation of the line if PQ. PE = 20.
17) If the line y - √3 x +3=0 cuts the parabola y²= x + 2 at A and B, then find the value of PA. PB (where P=(√3,0).
18) If x + 4y -5=0 and 4x + ky +7=0 are two perpendicular lines then k is
a) 3 b) 4 c) -1 d) -4.
19) A line L passes through the points (1,1) and (2,0) and another line M which is perpendicular to L passes through the point (1/2,0). The area of the triangle formed by these lines with y-axis is
a) 25/8 b) 25/16 c) 25/4 d) 25/32.
20) If the straight line 3x + 4y+ 5 - k(x + y +3)= 0 is parallel to y-axis, then the value of k is
a) 1 b) 2 c) 3 d) 4
21) If the algebraic sum of perpendiculars from n given points on a variable straight line is zero then show that the variable straight line passes through a fixed point.
22) Show that no line can be drawn through the point (4,-5) so that its distance from (-2,3) will be equal to 12.
23) Three lines x+ 2y+3=0, x + y= 7, 2x - y= 4 form 3 sides of two squares. Find the equation of remaining sides of these squares.
24) Find the equation to the sides of an isosceles right angled triangle, the equation of whose hypotenuse is 3x + 4y= 4 and the opposite vertex is the point (2,2).
25) Let P(sinθ, cosθ)(0≤θ≤2π) be a point and let OAB be a triangle with vertices (0,0), ((√3/2),0) and (0,√(3/2)). Find θ if P lies inside the ∆ OAB.
26) Through what angles should the axes be rotated so that the equation 9x² - 2√3xy = 10 may be changed to, 3x² + 5y²= 5?
27) For the straight lines 4x + 3y= 6, 5x +12y +9= 0, find the equation of the
a) bisector of the obtuse angle between them.
b) bisector of the acute angle between them.
c) bisector of the angle which contains origin.
28) Show that each member of the family of straight lines
(3sinθ + 4 cosθ)x + (2 sinθ - 7 cosθ)y + (sinθ + 2 cosθ)= 0 (θ is a parameter) passes through a fixed point.
29) λx²- 10xy + 12y²+ 5x - 16y -3=0 represents a pair of straight lines, then λ is equal to
a) 4 b) 3 c) 2 d) 1.
30) Show that the two straight lines x²(tan²θ+ cos²θ) - 2xy tanθ + y² sin²θ = 0 represented by the equation are such that the difference of their slopes is 2.
31) If pair of straight lines x¹- 2pxy - y²= 0 and, x² - 2qxy - y²= 0 be such that each pair bisects the angle between the other pair, show that pq= -1.
32) The chord √6y = √8 px + √2 of the curve py²+ 1= 4x subtends a right angle at origin then find the value of p.
EXERCISE - A
1) Find the cartesian co-ordinates of the point whose polar coordinates are
a) (5, π- tan⁻¹(4/3)).
b) 5√2, π/4).
2) Find the polar coordinates of the points whose cartesian co-ordinates are
a)!(-2,-2).
b) (-3,4).
3) Transform the equation r²= a² cos2θ into cartesian form.
4) transform the equation x²+ y²= ax into polar form.
OBJECTIVE - A
1) The polar coordinates of the point whose cartesian coordinates are (- 1, - 1) is :
a) (√2,π/4) b) (√2, 3π/4) c) (√2, - π/4) d) (√2, -3π/4)
2) The cartesian co-ordinates of the point whose polar coordinates are (13, π - tan⁻¹(5/12)) is:
a) (12,5) b) (-12,5) c) (- 12,-5) d) (12, - 5)
3) The transformation equation of r² cos²θ = a² cos2θ to cartesian form is (x²+ y²)x²= a²λ, then the value of λ is:
a) y²- x² b) x²- y² c) xy d) x²y²
4) The coordinate of P' in the figure is:
a) (3, π/3) b) (3, -π/3) c) (-3, -π/3) d) (-3, π/3)
5) The cartesian coordinates of the point Q in the figure is:
a) (√3,1) b) (-√3,1) c) (-√3,-1) d) (√3, -1)
SUBJECTIVE - A
1) A point lies on x-axis at a distance 5 units from y-axis. What are the coordinates ?
2) A point lies on y-axis at a distance of 4 units from x-axis. What are its coordinates ?
3) A point line on negative direction of x-axis at a distance 6 units from y-axis. What are its coordinates ?
4) Transform the equation y = x tanβ to polar form .
5) Transformation the equation r= 2β cosθ to cartesian form.
EXERCISE - B(1)
1) Prove that the distance of the point (a cosβ, a sinβ) from the origin is independent of β.
2) Find the distance between the points (a cosβ, a sinβ) and (a cosγ, a sinγ) where a >0.
3) If the point (x, y) be equidistant from the points (6,-1) and (2, 3), prove that x - y = 3.
4) Using distance formula, show that the points (1,5), (2,4) and (3,3) are collinear .
5) An equilateral triangle has one vertex at the point (0,0) and another at (3, √3). Find the coordinanates of the third vertex.
6) Show that the four points (1,-2),(3,6),(5, 10) and (3,2) are the vertex of a parallelogram.
7) Let the opposite angular points of a square be (3,4) and (1, -1). Find the coordinates of the remaining angular points.
8) Find the circumcenter of the triangle whose vertices are (-2,3),(-1,0) and (7,-6). Also find the radius of the circumcircle.
9) if the line segment joining the point A(a, b) and B(c, d) subtends an angle θ at the origin O, prove that
Cosθ = (ac + bd)/√{(a²+ b²)(c²+ d²)}
EXERCISE - B(2)
1) Show that the triangle, the coordinates of whose vertices are given by integers, can never be an equilateral triangle.
2) In any triangle ABC, show that
AB²+ AC²= 2(AD²+ BD²) Where D is the middle point of BC.
3) Let ABCD be a rectangle and P be any point in its plane.
Show that PA²+ PC²= PB²+ PD².
Distance between two points in polar coordinates
EXERCISE - B(3)
1) Prove that the points (0,0), (3, π/2) and (3, π/6) are the vertices of an equilateral triangle.
OBJECTIVE - B
1) If the distance between the point (a, 2) and (3,4) be 8, then a =
a) 2+ 3√15 b) 2- 3√15 c) 2± 3√15 d) 3 ± 2√15
2) The three points (-2,2), (8,-2) and (-4,-3) are the vertices of:
a) an isosceles triangle
b) an equilateral triangle
c) a right angle triangle d) none
3) The distance between the point (3, π/4) and (7,5π/4) is:
a) 8 b) 10 c) 12 d) 14
4) Let A(6, -1), B(1,3) and C(x,8) be three points such that AB= BC then the value of x are:
a) 3, 5 b) -3, 5 c) 3, -5 d) -3, -5
5) The points (a+1,1),(2a+1,3) and (2a+2, 2a) are collinear, if:
a) a= -1,2 b) a= 1/2 ,2 c) a= 2,1 d) a= -1/2,2
6) If A= (3,4) and B is a variable point in the line |x|= 6. If AB ≤ 4 then the number of positions of B with integral coordinanates is
a) 5 b) 6 c) 10 d) 12
7) The number of points on x-axis which are at a distance c units (c < 3) from (2,3) is :
a) 1 b) 2 c) 0 d) 3
8) The points on the axis of y which its equidistant from (-1, 2) and (3,4) is:
a) (0,3) b) (0,4) c) (0,5) d) (0,-6)
SUBJECTIVE -B
1) Find the distance between the points (at₁², 2at₁) and (at₂², 2at₂), where t₁, and t₂ are the roots of the equation x²- 2√3 x + 2 =0 and a> 0.
2) If P(at², 2at), Q(a/t², 2a/t) and S(a,0) be any the three points , show that 1/SP + 1/SQ is independent of t.
3) Prove that the points (3,4), (8,-6) and (13,9) are the vertices of a right angled triangle.
4) show that the points (0,-1),(6,7), (-2,3) and (8,3) are the vertices of a rectangle.
5) Find the circumcentre and circumradius of the triangle whose vertices are (-2,3),(2,-1) and (4,0).
6) The vertices of a triangle are A(1,1), B(4,5) and C(6,13). Find cos A.
7) Two opposite vertices of a square are (2,6) and (0,-2). Find the coordinanates of the other vertices.
8) If the point (x,y) is equidistant from the points (a+ b), b- a) and (a- b, a+ b), prove that bx = ay.
9) if a and b are real numbers between 0 and 1 such that the points (a,1),(1, b) and (0, 0) form an equilateral triangle, find a and b.
10) An equilateral triangle has one vertex at (3,4) and another at (-2,3). Find the coordinanates of the third vertex.
11) If P be any point in the plain of square ABCD, prove that PA²+ PC²= PB²+ PD².
SECTION FORMULA
EXERCISE - C
1) Find the coordinates of the point which divides the line segment joining the points (5, -2) and (9,6) in the ratio 3:1.
2) Find the length of median through A of a triangle whose vertices are A(-1,3), B(1,-1) and C(5,1)
3) Determine the ratio in which y - x + 2= 0 divides the line joining (3, -1) and (8, 9).
4) The coordinanates of three consecutive vertices of a parallelogram are (1,3), (-1, 2) and (2,5). Then find the coordinates of the fourth vertex.
5) In water ratio does x-axis divide the line segment joining (2,-3) and (5,6)?
6) The mid point of the sides of a triangle are (1,2),(0 ,1) and (2,-1). Find the coordinanates of the vertices of a triangle with the help of two unknowns.
7) Prove that in a right angled triangle the mid point of the hypotenuse is equidistant from its vertices.
8) Show that the line joining the midpoints of any sides of a triangle is half the third side.
EXERCISE - D
1) Find the coordinates of a point which divides the externally the line joining (1,-3) and (-3,9) in the ratio 1 :3.
2) The line segment joining A(6,3) to B(- 1,-4) is doubled in the length by having its length added to each end. Find the coordinates of the new events.
3) Using section formula show that the points (1, -1), (2,1) at(4, 5) are collinear .
4) Find the ratio in which the point (2, y) divide the line segment joining (4,3) and (6, 3) and henr find the value of y.
EXERCISE - E
1) Find the harmonic conjugate of the point R(5,1) with respect to the points P(2,10) and Q(6, -2).
EXERCISE - F
1) Two vertices of a triangle are (-1,4) and (5,2). If its centroid is (0,-3), find the third vertex.
2) The vertices of a triangle are (1, 2), ( h, - 3) and (-4, k).
Find the value of √{(h +k)²+ (h + 3k)²}.
If the centroid of the triangle be at the point (5,-1).
3) If D(-2,3), E(4,-3) and F(4,5) are the midpoint of the sides BC, CA and AB of triangle ABC, then find √{|AG|²+ |BG|² - |CG|²} where G is the centroid of ∆ ABC .
4) If G be the centroid of the triangle ∆ABC and O be any other point in the plane of the triangle ABC, then show that OA²+ OB²+ OC²= GA²+ GB²+ GC²+ 3GO².
5) If G be the centroid of ∆ ABC, show that AB²+ BC²+ CA²= 3(GA²+ GB²+ GC²).
6) The vertices of a triangle (1, a), (2,b) and (c², -3)
a) Show that its centroid can not lie on the y-axis.
b) Find the condition that the centroid may lie on the x-axis.
EXERCISE - G
1) Find the co-ordinate the incentre of the triangle is vertices are (4,-2),(-2,4) and (5,5).
2) If (3/2, 0),(3/2, 6) and (-1,6) are mid points of the sides of a triangle, then find
a) Centroid of the triangle.
b) Incentre of the triangle.
EXERCISE - H
1) If a vertex of a triangle be (1,1) and the middle points of two sides through it be (-2, 3) and (5,2) then find the centroid and the incentre of the triangle. {(5√2- 5√17+ 9√13)/(5√2+ √17+ √13), (5√2+ 5√17+ 3√13)/(5√2+ √17+ √13)}
2) If G be the centroid and I be the incentre of the triangle with vertices A(-36, 7), B(20,7) and C(0,-8) and GI= (25√205)/3λ, then find the value of λ. 1/25
3) In a triangle ABC with vertices A(1,2), B(2,3) and C(3,1) and angle A= cos⁻¹(4/5), angle B= angle C = cos⁻¹(1/√10), then find the circumcentre of the triangle ABC .
4) if the co-ordinate of the midpoint of the sides of a triangle are (1,1), (2,-3) and (3,4) then find the excentre opposite to the vertex A.
5) If a triangle has its orthocentre at (1,1) and circumcentre at (3/2,3/4) then find the centroid and nine point centre. (4/3,5/6), (5/4,7/8)
6) The vertices of a triangle are A(a, a tan α), B(b, b tanβ) and C(c, c tanγ). If the circumcentre of ∆ ABC coincides with the origin and H(bar x, bar y) is the orthocentre, then show that
bar y/bar x = (sinα + sinβ + sinγ)/(cosα + cosβ + cosγ).
EXERCISE - I
1) The co-ordinates of ABC are (6,3),(-3,5) and (4,-2) respectively and P is any point (x,y). Show that the ratio of the areas of the triangle PBC and ABC is |(x+ y - 2)|/7.
2) Find the area of the Pentagon whose vertices are A(1,1), B(7, 21), C( 7, -3), D(12,2) and E(0,-3).
3) Show that the point (a,0), (0,b) and (1,1) are collinear , if 1/a + 1/b =1
4) Prove that the co-ordinates of the vertices of an equilateral triangle can not all be rational.
5) The value coordinanates of two points A and B are (3,4) and (5,-2) respectively. Find the Co-ordinates of any point P if PA= PB and area of ∆ APB is 10.
6) Find the area of the triangle formed by the straight line 7x - 2y + 10 = 0, 7x + 2y -10= 0 and 9x + y +2=0 (without solving the vertices of the triangle).
7) If ∆₁ is the area of the triangle with vertices (0,0), (a tanα, b cotα), (asinα, b cosα); ∆₂ is the area of the triangle with vertices (a,b), (a sec²α, b cosec²α), (a+ a sin²α, b + b cos²α) and ∆₃ is the area of the triangle with vertices (0,0), (a tanα, - b cotα), (a sinα, b cosα). Show that there is no value of α for which ∆₁ , ∆₂ and ∆₃ are in GP.
MISCELLANEOUS EXERCISE- A
OBJECTIVE - A
1) The coordinanates of the middle points of the sides of the triangle are (4,2),(3,3) and (2,2), then the coordinates the centroids are
2) The incentre of the triangle whose vertices are (-36,7),(20,7) and (0,-8) is:
a) (0,-1) b) (-1,0) c) (1,1) d) (1/2,1)
3) If the orthocentre and centroid of a triangle are (-3,5) and (3,3) then its circumcentre is:
a) (6,2) b) (3,-1) c) (-3,5) d) (-3,1)
4) An equilateral triangle has each side to a. If the co-ordinates of its vertices are (x₁, y₁), (x₂,y₂), and (x₃, y₃) then the square of the determinants
x₁ y₁ 1
x₂ y₂ 1
x₃ y₃ 1 equals to
a) 3a⁴ b) 3a⁴/2 c) 3a⁴/4 d) 3a⁴
5) The vertices of a triangle are A(0,0), B(0,2) and C(2,0). The distance between circumcentre and orthocentre is:
a) √2 b) 1/√2 c) 2 d) 1/2
6) A(a,b), B(x₁, y₁), and C(x₂,y₂) are the vertices of the triangle. If a, x₁, x₂ are in GP with common ratio r , and b, y₁, y₂ are in GP with common ratio s, then area of ∆ ABC is:
a) ab(r-1)(s-1)(s- r)
b) (ab/2) (r+1)(s+1)(s- r)
c) (ab/2) (r-1)(s-1)(s- r)
d) ab(r+1)(s+1)(r - s)
7) The point (x+1, 2),(1, x+2), (1/(x+1), 2/(x+1)) are collinear then x is equals to:
a) -4 b) -8 c) 4 d) 8
8) The vertices of a triangle are (6,0),(0,6) and (6,6), then distance between its circumcentre and centroid, is:
a) 2√2 b) 2 c) √2 d) 1
9) The nine point centre of the triangle with vertices (1,√3),(0,0) and (2,0) is :
a) (1,√3/2) b) (2/3, 1/√3) c) (2/3, √3/2) d) (1,1/√3)
19) The vertices of a triangle are (0,0),(1,0) and (0,1). Then ex-centre opposite to (0,0) is:
a) (1- 1/√2, 1+ 1/√2)
b) (1+ 1/√2, 1+ 1/√2)
c) (1- 1/√2, 1+ 1/√2)
d) (1- 1/√2, 1- 1/√2)
MISCELLANEOUS SUBJECTIVE - A
1) α, β, γ are the real roots of the equation x³- 3px²+ 3qx -1= 0 then find the centroid of the triangle whose vertices are (α, 1/α), (β,1/β) and (γ,1/γ).
2) If centroid of a triangle be (1,4) and the Co-ordinates of its any two vertices are (4,-8) and (-9,7), find the area of the triangle.
3) Find the centroid and incentre of the triangle whose vertices are (1,2),(2,3) and (3,4).
4) Show that the area of the triangle with vertices (λ, λ-2),(λ+3, λ) and (λ+2, λ+2) is independent of λ.
5) Prove that the points (a, b+ c), (b, c + a) and (c, a+ b) are collinear.
6) Show that the points (a, b) and (a - c, b - d) are collinear, if ad = bc.
7) If the points (x₁,y₁),(x₂,y₂) and (x₃,y₃) are collinear, show that
∑{(y₁ - y₂)/(x₁x₂)}= 0, i.e., (y₁ - y₂)/x₁x₂ + (y₂ - y₃)/x₂x₃ + (y₃ - y₁)/x₃x₁ = 0.
8) The coordinanates of points A, B, C and D are (-3,5),(4,-2),(x, 3x) and (6,3) respectively and ∆ ABC/∆ BCD = 2/3, find x.
9) Find the area of the hexagon whose vertices taken in order are (5,0),(4,2),(1,3),(-2,2), (-3,-1) and (0,-4).
MISCELLANEOUS - B
1) Find the polar coordinates of the point whose cartesian coordinates are (-√3,1). (2,5π/6)
2) Find the cartesian coordinates of the point whose polar coordinates are (√2, 5π/4). (-1,-1
3) Express r= 2a cosθ in cartesian form. x²+ y²= 2ar.
4) Express x²- y²= 2ax in polar coordinates form.
5) Express the following relations in polar coordinates:
a) y= x tanα (α is a constant). sin( θ - α)= 0
b) y²= 4x +3. r² sin²θ = 4r cos θ +3
c) x²+ y²= a². r= a
d) xy= 9. r² sin2θ = 18
e) 4x²+ 3y²= 12. 1/r²= cos²(θ/3) + sin²(θ/4)
f) 2x - 3y = 8. 2 cosθ - 3 sinθ = 8/r
g) x= 3. Cosθ = 3/r
h) x²+ y²= 2ax. r= 2a cosθ
i) x²+ y²- 2x + 4y =0. r - 2 cosθ + 4 sinθ = 0
6) Express the following relations in cartesian co-ordinates:
a) r= 2a cosθ. x²+ y²- 2ax
b) r² - 6r cosθ +5= 0. x²+ y²- 6x +5=0
c) r= 5. x²+ y²= 25
d) r²= 4/(cos²θ - sin²θ). x²- y²= 4
e) l/r = 1+ e cosθ; (l, e are constant). x²+ y²= (ex - l)²
f) r= a sinθ . x²+ y²= ay
7) Find the lengths of the sides of the triangle whose vertices are the points (-2,3), (-2,-1), (4,-1). 4, 6, 2√13, right angle at B
8) Show that the following points lie on a straight line (-3,-2),(5,2),(9,4).
9) Prove that the distance between the point (at², 2at) and (a/t², -2a/t) is a(t + 1/t)².
10) prove that the distance between the point (a cosθ , a sinθ ) and (acosβ, a sinβ) is 2a sin{(θ-β)/2}. (If θ> β).
11) Find the distance between two points P₁ and P₂ whose polar coordinates are respectively:
a) (2, 30°) and (4 0,120°). 2√5
b) (a,π/2) and (3a, π/6). a√7
12) Prove that the points (0,0),(3,π/2) and (3,π/6) form an equilateral triangle.
13) Show that the quadrilateral with the vertices P₁: (-3/2,4), P₂: (-7/2,3), P₃: (1,0), P₄: (3,1) is a parallelogram.
14) The segment joining P₁: (1,3) and P₂: (5,-2) is trisected . Find the point of trisection P near to P₁.
15) The segment from P₁:(5,-4) to P₂: (7,-9) is extended beyond P₂ so that its length is doubled. Find the coordinates of the terminal point P. (9,-14)
16) A circle with centre at A(-4,1) has one end of a diameter at B(2,6). Determine the coordinanates C(x,y) of the other and. (-10,-4)
17) If the point (9,2) divides the segment of the line from A(6,8) to B(a,b) in the ratio 3:7, find the co-ordinate of B. (16,-12)
18) Determine the coordinates of the vertices of a triangle if the middle points of its sides are (3,2),(-1,-2) and (5,-4). (-3,4),(9,0),(1,-8)
19) Show that analytically that the lines joining the middle points of the adjacent sides of the quadrilateral A(-3,2), B(5,4), C(7,-6), D(-5,-4) form a second quadrilateral whose perimeter is equal to the sum of the diagonals of the first.
20) Find the area of a triangle with vertices are (3,1),(2k, 3k), (k, 2k) and show that the distinct points are collinear when k= -2.
21) Find the area of the triangle whose vertices are (a, b+ c),(b, c+ a), and (c, a+ b). 0
22) Show that the line joining the middle points of the sides of the triangle (2,-3),(4,2),(-5,-2), divide the triangle and four triangles whose areas are equal .
23) Show that area of the triangle with the vertices (t, t-2),(t+3, t),(t+2, t+2) is independent of t.
24) A, B, C, D are the points (3,1),(2,4),(2,2), (3- 2t, t²+ 5) and O is the origin 0,0); if area of ∆OAB = ∆OCD, (both in magnitude and in sign), find the possible values of t. 1 or -3
25) The vertices of a quadrilateral in order, are (-2,3),(-3,-2), (2,-1),(x,y); if its area is 14. Show that x+ y = 2.
26) Show that the point (-1,-2) is the centre of a circle passing through the points (11,3),(-1,-15),(-13,-7) and (4, -14(.
27) A, B are points (-8,0),(-2,0): a point P(x,y) in such that|bar PA|= 2 |bar PB|. Show that x²+ y²= 16.
28) A line is of length 10 units . Its one end is at the point (2,-3); if the abscissa of the other end be 10. Prove its ordinate is either 3 or -9.
29) if the coordinates of the three vertices triangle are (-2,5),(- 4,-3) and (6,-2), find the coordinates of the centroid of the triangle. (0,0)
30) The co-ordinates of the vertices of a triangle are (4,-3),(-5,2) and (x,y). If the centre of gravity of the triangle is at the origin, find x, y. 1,1
31) The points (1,2,),(2,4),(t,6) are collinear, find t. 3
32) If the three points (a,0),(0,b) ay(2,2) are collinear, prove that 1/a + 1/b = 1/2.
33) What are the coordinates of P if O be the origin and Q(-2,-4) is the point on OP such that OQ= (1/3) OP ? -6,12)
34) The coordinates of the vertices of a triangle are (0,0),(5,3) and (3,5) respectively, find the circumcentre and circumradius of the triangle. (17/8,17/8), 17√2/8
35) Triangle OAB has vertices (0,0),(b cosθ, - b sinθ) and (sinβ, cosβ). Show that area of the triangle OAB is maximum when θ = β and find the maximum area. b/2
36) The centre of a circle is at (5,3) and its radius is 5. Find the length of the chord which is bisected at (3,2). 4√5
37) A circle with centre (0,-13) has the axis of x as one of its tangents. Does the circle pass through (11,-6) ? Through (-5,-1)? No, yes
38) If the three points (a,b),(a+ k cosθ, b + k sinθ) and (a+ k cosβ, b + k sinβ) are the vertices of an equilateral triangle, then which of the following results is true and why ?
a) |α -β|=π/4
b) |α - β|= π/2
c) |α- β|=π/6
d) |α - β|=π/3.
MISCELLANEOUS - C
1) Show. that the centroid of the triangle ABC were A(-2,5), B(-4,-3), C(6,-2) is the origin.
2) Find the area of the triangle ABC with vertices A(x₁, y₁), B(x₂,y₂) and C(x₃, y₃). If ABCD be collinear , is it, in general , true that the three points have formed a triangle of zero area ?
3) Find the conditions that the three points A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be collinear.
4) show that area of the triangle of the vertices (-5,-2),(2,2) and (3,4) will be 5 square units.
5) Show that (1,5),(3,14) and (-1,-4) are collinear.
6) Show that the points (2,0),(0,2),(√3+1, √3+1) are the vertices of an equilateral triangle.
7) Prove that the triangle whose vertices at the points (1,8),(3,2),(9,4) is an isosceles right angled triangle.
8) Show that the points (-2,-1), (1,0),(4,3) and (1,2) are the vertices of a parallelogram. Is the Parallelogram is rectangle ?
9) Show that the point (-7,1),(5,-4),(10,8) and (-2,13) the vertices of a square and find its area.
10) Prove that the points (2,-2),(8,4),(5,7) and (-1,1) are the angular points of rectangle.
11) Prove that the points (2,5),(5,9), (9,12) and (6,8) when joined in order form a rambus.
12) Prove that the point (-1/14,39/14) is the circumcentre of a triangle whose vertices are (1,4),(2,3) and (-2,2). What is the length of the circum-radius ?
12) A circle with centre (3,2) passes through (13,- 10). Does this circle pass through (-11, 9)?
13) A circle with centre (0,13) has the axis of x as one of its tangents.. Does this circle pass through (11,- 6)? Through (-5,1)?
14) The centre of a circle is at (5,3) and its radius is 5. Find the length of the chord that is bisected at (3,2). 4√5
15) Find the radius of a circle with centre at (1,1), if a chord of length 10 units is bisected at (2,0). 3√3
16) A(1,2), B(3,4). A point P divides AB in the ratio 3:1 internally show that P=(13/7,20/7).
17) A line segment directed from (-3,2) to (1,-4) is trebled. Find the coordinates of the terminal point. (9,-16)
18) The segment joining P(1,3) and Q(5,-2) is trisected. Find the point of trisection B nearer to Q. (7/3,4/3)
19) Find the area of the triangle with vertices A(3,1), B( 2K, 3K) C(K, 2K) and show that the three distinct points A, B, C are collinear when k=-2.
20) Show that the area of the triangle formed by the points (-3,4),(6,2) and (4,-3) is 24.5 square units.
21) The points (2,3/2),(-3,-7/2),(t, 9/2)are collinear ; find t. 5
22) Find the area of the triangle with vertices (2,-1),(a+1, a-3), (a+2,a) and show that they are collinear if a = 1/2
23) If A(1,5) and B(-4,7), find the point P which divides AB in the ratio 2:3 internally . (-1,29/5)
24) What are the coordinates of P if O be the origin and Q(-2,4) is the point on OP such that OQ= (1/3) OP ? (-6,12)
25) A(1,2), B(3,4) are two fixed points. P divides AB internally in the ratio 1:m. Show that P= {(3+m)/(1+m), (4+2m)/(1+m)}
26) a point divides internally the line segment joining (8,9) and (-7,4) in the ratio 2:3. Show that the coordinates of the points are (2,7).
27) Find the coordinates of the centroid of the triangle whose vertices are (0,0),(6,4),(4,6). (10/3,10/3)
28) Show that the centroid of the triangle with vertices at (4,-1),( 0,3) and (-4,-2) is the origin of the coordinates.
29) The coordinates of the vertices of a triangle are (4,-3),(- 5,2) and (x,y). if the centroid of the triangle is at the origin, find x and y. 1, 1
30) Find the area of the triangle having vertices at (1,4), (-1,2) and (-4,-1) interpret the results.
31) Show that the area the triangle having vertices at (a, 1/a),(b, 1/2), (c, 1/c) is {(b-c)(c- a)(a- b)}/2abc.
32) The point (1,2),( 2,4),(t,6) are collinear, find t. 3
33) if the points (a,0),(0,b) and 2,2) are collinear, prove that 1/a + 1/b = 1/2
34) If (a,b), (A', b'), (a- A', b - b') be collinear, prove that ab'= a'b.
35) Find the condition that the points (a,b),(b,a) and (a², - b²) are in straight line. a³+ b³- a²b - ab²- a²+ b²= 0
36) Show that the point (-1,-2) is the centre of a circle passing through the points (11,3),(- 1,15),(-13,-7) and (4,-14).
37) A, B are points (-8,0),(-2,0); a point P(x,y) is such that |PA|= 2|PB|. Show that x²+ y²= 16
38) a line is of length 10 units. Its one end is at (2,-3); if the abscissa of the other end be 10, prove that its ordinate is either 3 or -9.
39) The coordinates of the vertices of a triangle (0,0)(5,3) and (3,5) respectively, find the circumcentre and the circumradius of the triangle triangle. 17/8,17/8,17√2/8
40) Triangle OAB has vertices (0,0),(b cosx , - b sinx) and (siny, cosy). Show the area of the triangle OAB is maximum when x= y ay find the maximum area. b/2
41) if the three points (a,b),(a+ k cosx, b+ k sinx) and (a+ k cosy, b + k siny) are the vertices of an equilateral triangle, then which of the following results is true
a) |x - y|= π/4
b) |x - y|= π/2
c) |x - y|= π/6
d) |x - y|= π/3
42) Find the coordinates of the points which divide, internal and externally, the line joining the point (a+ b, a- b) to the point (a- b, a+ b) in the ratio a: b (0< b < a). {(a²+ b²)/(a+ b), (a²+2ab- b²)/(a+b)}; {(a²- 2ab- b²)/(a- b), (a²+ b²)/(a-b)}
53) If (3,-1),(-4,-3) and (1,5) are the three vertices of parallelogram and the 4th vertex lies in the first quadrant, find the coordinates of the fourth vertex. (8,7)
54) a point with abscissa 6 lies on the lines joining the two points (2,5),(8,2). Find the ordinatr. 3
55) if O be the origin and if the coordinates of anr two points X and Y be respectively (a,b) and (c,d) show that OX . OY cos XPY= ac+ bs.
56) Find the area of the triangles the coordinators of whose vertices are:
a) (am₁², 2am₁),(am₂², 2am₂) and (am₃², 2am₃). a²(m₁- m₂)(m₂- m₃)(m₃- m₁)
b) (a cosx₁, b sinx₁),(acosx₂, b sinx₂) at(a sinx₃, b sinx₃). 2ab sin(x₂- x₃)/2 . sin(x₃ - x₁)/2 . sin(x₁ - x₂)/2
57) three vertices of a parallelogram are (2,1); (4,-5),( 4,2). Find the area of the parallelogram and also fourth vertex. 14; (6,-4)
58) A, B, C are points (x,y),(- 3,2),(-4,-4). If the area of ∆ ABC is 35/2, show that 6x - y - 15=0.
59) The lines joining the midpoints of opposite sides of a quadrilateral bisect each other.
60) The diagonals of a parallelogram bisect each other.
61) The midpoint of the hypotenuse of a right angled triangle is equidistant from the three vertices.
62) if two medians of a triangle are equal show that the triangle is isosceles .
63) An isosceles triangle has two equal medians.
64) The sum of the squares of the distances of any point in the plane of a given rectangle to two opposite vertices equals the sum of the squares of the distances from it to the two other vertices.
65) Verify In any triangle ABC, AB²+ AC²= 2(AD²+ DC²)= 2(AD²+ BD²) where D is midpoint of BC.
66) If G be the centroid of a triangle ABC and P any other point on the same plane of the triangle, then 3(GA²+ GB²+ GC²)= (BC²+ CA²+ AB² and PA²+ PB²+ PC²= GA²+ GB²+ GC²+ 3GP²
67) The lines joining the middle points of opposite sides of a quadrilateral and the line joining the middle points of its diagonals meet in a point and bisect one other .
68) The line joining the midpoints of the two sides of a triangle of a triangle is half the third side.
69) If P be the point which divides the line segment XY internally in the ratio m: n and Q be a point not lying on the line joining X and Y, then
Area of ∆ QPX/area of ∆QPY = m/n
70) A:(3,0), B:(0,6) and C:(6,9) from a triangle ABC. A line cuts AB and AC at D and E respectively, so that D divides AB in the ratio 1:2 and E divides AC also in the same ratio. Prove that
The numerical measure of ∆ ABC/the numerical measure of ∆ ADE = 9/1.
71) Show that in two ways that three points (1,5),( 3,14) and (-1,-4) lie on a straight line.
72) A(6,3), B(-3,5), C(4,-2) and P(a,b) are four points on a plane. Prove that the numerical measure of the triangle PBC and ABC and the ratio (a+ b -2): 7.
73) The line segment joining A(b cosx, b sinx) and B (a cosy, a siny) is produced along AB to same point C(x,y) so that AC: CB = b : - a(a,b >0); prove that
y+ x cot{(x+ y)/2}= 0
LOCUS AND ITS EQUATION
Locus: The locus of a moving point is the path traced out by that point under one or more given conditions .
Equation of a Locus
A relation f(x,y)= 0 between x and y which is satisfied by each point on the locus and such that each point satisfying the equation is on the locus is called the equation of the Locus.
EXERCISE - A
1) Find the locus of a point which moves such that its distance from the point (0, 0) is twice its distance from the y-axis .
2) Find the locus of the moving point P such that 2PA = 3PB, where A (0, 0) and B is (4, -3).
3) A point moves so that the sum of the squares of its distance from two fixed points A(a,0) and B (-a,0) is constant and equal to 2c², find the locus of the point a point.
4) A point moves such that the sum of it distance from two fixed points (ae,0) and (-ae,0) is always 2a. Prove that the equation of the locus is x²/a² + y²/b²= 1 where b²= a²(1- e²)
5) Find the equation of the locus of a point which moves so that the difference of its distances from the points (3,0) and (-3,0) is 4 units.
6) The ends of the hypotenuse of a right angled triangle are (6,0) and (0,6). Find the locus of the third vertex.
7) Find the equation of the locus of a point which moves so that the sum of their distances from (3,0) and (-3, 0) is less than 9.
8) Find the locus of a point whose coordinate are given by x= t + t², y= 2t +1, where t is variable .
9) A stick of length L rests against the floor and a wall of a room. If the stick begins to slide on the floor, find the locus of its middle point.
10) Find the locus of the point of intersection of the lines x cos β + y sin β = a and x sin β- y cos β= b where β is variable.
11) A variable lines cuts x-axis at A, y-axis at B where OA= a, OB= b (O as origin) such that a²+ b²=1.
Find the locus of
a) centroid of ∆ OAB.
b) circumcenter of ∆ OAB.
12) Two points P and Q are given, R is a variable point on one side of the line PQ such that angle RPQ - Angle RQP is a positive constant 2β . Find the locus of the point R.
CHANGES OF AXES OR THE TRANSFORMATIONS OF AXES
EXERCISE - B
1) Find the equation of the curve 2x²+ y²- 3x + 5y - 8= 0, when the origin is transferred to the point (-1,2) without changing the direction of axas.
2) The equation of a curve referred to the new axes, axes retaining their direction and origin is (4,5) is x²+ y² = 36. Find the equation referred to the original axes .
3) Shift the origin to a suitable point so that the equation y² + 4y + 8x - 2 = 0 will not contain term in y and the constant.
4) At what point the origin be shifted , if the coordinates of a point (-1,8) become (-7,3) ?
EXERCISE - C
1) If the axes are turned through 45°, find the transformed form of the equation 3x²+ 3y²+ 2xy = 2.
2) Prove that if the axes be turned through π/4 the equation x²- y²= a² is transformed to the form xy = λ. Find the value of λ.
3) Through what angle should the axes be rotated so that equation 9x² - 2√3 xy + 7y²= 10 may be changed to 3x²+ 5y²= 5 ?
4) If (x, y) and (X, Y) be the coordinates of the same points referred to two sets of rectangular axes with the same origin and if ux + by, when u and v are independent of x and y become VX + UY, show that u²+ v²= U²+ V²
DOUBLE TRANSFORMATION (ORIGIN SHIFTED AND AXES ROTATED)
EXERCISE - D
1) What does the equation 2x²+ 4xy - 5y² + 20x - 22y - 14 = 0
becomes when referred to rectangular axes through the point (-2,-3), the new axes being inclined at an angle of 45° with the old ?
2) Find λ if (λ, λ +1) is an interior point of ∆ ABC where A≅ (0,3); B(-2,0) and C≅ (6,1).
OBJECTIVE - 1
1) The equation of the locus of the points equidistant from (-1,-1) and (4,2) is:
a) 3x - 5y -7=0 b) 5x +3y -9=0 c) 4x +3y +2=0 d) x - 3y +5=0
2) The equation of the locus of a point which moves so that its distance from the point (ak, 0) is k times its distance from the point (a/k, 0), (k≠ 1) is :
a) x² - y² =a² b) 2x² - y² = 2a² c) xy = a² d) x² + y² =a²
3) if the coordinates of a variable point P be (t + 1/t, t - 1/t) where t is the variable quantity, then the locus of P is :
a) xy= 8 b) 2x² - y² =8 c) x² - y² =4 d) 2x² + 3y² =5
4) If the coordinates of a variable point P be (cos θ+ sinθ, sin θ - cos θ), where θ is the parameter, then the locus of P is:
a) x² - y² =4 b) x² + y² =2 c) xy= 3 d) x² + 2y² =3
5) If a point moves such that twice its distance from the axis of x exceeds its distance from the axis of y by 2, then its locus is:
a) x - 2y= 2 b) x + 2y= 2 c) 2y - x = 2 d) 2y - 3x = 5
6) The equation 4xy - 3x²= a² become when the axes are turned through an angle tan⁻¹2 is:
a) x² + 4y² =a² b) x² -4y² =a² c) 4x² +y² =a² d) 4x² - y² =a²
7) Transform the equation x² -3xy + 11x - 12y +36 =0 to parallel axes through the point (-4,1) becomes ax² + bxy +1=0 then b²- a =
a) 1/4 b) 1/16 c) 1/64 d) 1/256
SUBJECTIVE - 1
1) find the equation of the locus of all points equidistant from the point (2,4) and the y- axis.
2) Find the equation of the locus of the point twice as far from (-a,0) as from (a,0).
3) OA and OB are two perpendicular straight lines. A straight line AB is drawn in such a manner that OA+ OB = 8. Find the locus of the mid point of AB.
4) The ends of a length l move on two mutually perpendicular lines. Find the locus of the point on the rod which divides it in the ratio 1:2.
5) The coordinanates of three O, A, B are (0,0),(0,4) and (6,0) respectively. A point P moves so that area of ∆ POA is always twice the area of ∆POB. Find the equation to both parts of the locus of P.
6) What does the equation (a- b)(x²+ y²)- 2abx =0 become, if the origin be moved to the point (ab/(a- b), 0)?
7) The equation x²+ 2xy +4=0 is transformed to the parallel axes through the point (6, λ). For what value of λ its new form passes through the new origin ?
8) Show that if the axes be turned through (15/2)°; the equation √3 x²+ (√3 -1)xy - y²= 0 become free of xy in its new form.
9) Find the angle through which the axes may be turned so that the equation Ax + By + C =0 may reduce to the form x= constant, and determine the value of this constant.
10) Transform 12x²+ 7xy - 12y²- 17x - 31y - 7 =0 to rectangular axes through the point (1,-1) inclined at an angle tan⁻¹(4/3) to the original axes.
MISCELLANEOUS - 2
1) A point moves in a plane such that its distance from (2,3) exceeds its distance from y axis by 2. Find the equation of the locus . y²-6y- 8x +9=0.
2) Find the locus of the point equidistant from x and y axis. y= ± x
3) A point moves in the xy-plane in such a way that its distance from the point (4,0) is always equal to its distance from the y-axis. Find the equation to the locus of the moving point. y² - 8x +16 =0.
4) A point moves in such a manners that the sum of the squares of the distance from it to the points (a,0) and (-a,0), is 2b². Find the locus of the point. x²+y²= b²- a²
5) Find the locus of the point which moves such that it forms a triangle of area 12 square units with (3,2) and (5,6). 2x - y =16 and 2x - y+8=0
6) Find the locus of a point whose distance from (2,4) and (1,-2) are in the ratio 2:3. 5x²+ 5y²-88y- 46x +205=0.
7) P is a variable point (t+3, 2t-1), when t may have any value. Find the locus to the locus of P. 2x - y=7
8) a point P(x, y) moves according to the law. Find the equation to the locus of following:
a) equidistant from (-4,-4) and (2,8). x+ 2y=3
b) Distance of P from (4,0) is two thirds of the distance from (9,0). x²+ y²=36
c) P is always at a distance 2 units from (-3,0). x²+ y²+ 6x +5=0.
d) distance of P from (0,5) is two -distance of its distance from the x-axis. 9x²+ 5y²- 90y +225=0
e) Distance of P from (3,2) is twice its distance from the y-axis. 3x²- y²+4y+ 6x 139=0
f) distance of P from (2,0) is equal to distance from y-axis . y²= 4(x -1)
g) distance of P from (2,3) exceeds its distance from y-axis by 2. y²-6y- 8x +9=0.
9) The sum of the squares of the distance of P from (0,a) and (0,-a) is 6a². x²+y² =2a²
10) The sum of the distances of P from (4,0) and (-4,0) is 10. 9x²+25y²=225
11) The difference of the distances of P from (2,0) and (-2,0) is 2. 3x²-y² -3=0
12) A(a,0) and B(-a,0) are two fixed points, obtain the equation to the locus of a moving point P where
a) PA²- PB²= 2k² (a constant quantity). 2ax + k²= 0
b) PA²= n. PB (n is a constant). (n²-1)(x²+ y²+ a²)+ 2ax(n²+1)=0
c) PB²+ PC²= 2PA², C being the point (e,0). (6a- 2c)x = a²- c².
13) Find the locus of the point equidistant from x and y axis. y= ± x
14) a point moves in the xy plane in such a way that its distance from (4,0) is always to its distance from the y-axis. Find the equation of the locus of the moving point. y²- 8x +16=0
15) A point moves in such a manner that the sum of the squares of the distances from it to the point (a,0) and (-a,0) is 2b². Find the locus of the point. x²+y²=b²- a²
16) it twice the abscissa of a point moving in the xy-plane always exceeds 3 times its ordinate by 1, show that the locus of the point is a straight line 2x= 3y +1.
17) A fixed point is at a perpendicular distance k from a fixed straight line and a point moves so that its distance from the fixed point is always equal to its distance from the fixed line. Prove that the equation to its locus is x²+ 2ky = k². The axes are to be chosen suitably .
18) Two points O(0,0) and A(3,4) are given. Find the equation the locus of B, if the area of ∆ OAB, the vertices being taken in this order, is 7 square units. 4x - 3y +14=0
19) Find the locus of a point which forms a triangle of area 21 square units with (2,-7) and (-4,3).
20) P is a variable point (t+3, 2t -1), where t may have any value. Find the equation to the locus of P. 2x - y -7=0
21) P is a variable point (at², 2at), where t may have any value, show that equation to the locus of P is y²= 4ax.
22) P is a variable point (a cosθ, b sinθ), where θ any assume any value, show that the equation to the locus of P is x²/a²+ y²/b²= 1.
23) A and B are two points having coordinanates (-5,3) and (2,4) respectively. Find the locus of a point P such that PA: PB = 3:2.
24) Find the locus of a point whose distance from (3,4) and (1,-2) are in ratio 2:3. 5x²+ 5y²- 76x - 48y +46=0
REMOVAL OF THE TERM xy FROM f(x,y)= ax²+ 2hxy + by² WITHOUT CHANGING THE ORIGIN
EXERCISE - E
1) Given the equation 4x²+ 2√3 xy + 2y²= 1, through what angle should the axes be rotated so that the term in xy be wanting from the transformed equation.
1) The four points A(α,0), B(β,0), C(γ,0) and D(δ,0) are such that α , β are the roots of the equation ax²+ 2hx + b=0 and γ, δ are those of the equation a'x²+ 2h'x + b'= 0. Show that the sum of the ratios in which C and D divide AB is zero, if ab' + a'b = 2hh'.
2) The circumcenter of a triangle with vertices (a, a tanα), B(b,b tanβ) and C(c, c tanγ) lies at the origin, where 0< α, β, γ < π/2 and α + β + γ =π. Show that its orthocentre lies on the line
4 cos(α/2) cos(β/2) cos(γ/2) x - 4 sin(α/2) sin(β/2) sin(γ/2) y = y
3) if m₁ and m₂ are the roots of the equation
x²+ (√3 +2)x + (√3-1)=0
Show that the area of the triangle formed by the lines y= m₁x, y= m₂x and y= c is c²(√33+√11)/4.
4) if x coordinanates of two points B and C are the roots of equation x²+ 4x +3= 0 and their y-cordinates are the roots of equation x²- x - 6 =0. If x co-ordinate of B is less than x coordinanates of C and y coordinanate of B is greater than the y coordinanate of C and coordinanates of a third point A be (3,-5), find the length of the bisector of the interior angle at A.
5) A line L intersects three sides BC, CA and AB of a triangle in P,Q,R respectively, show that
BP/PC . CQ/QA . AR/RB = -1.
6) The distance between two parallel lines is unity. A point P lies between the lines at a distance a from one of them. Find the length of a side of an equilateral triangle PQR, vertex Q of which lies on one of the parallel lines and vertex R lies on the other line.
7) In a ∆ ABC, A= (α , β), B(1,2), C=(2,3) and point A lies on the line y= 2x +3 where α , β ∈ I, if the area of ∆ ABC be such that [∆] = 2, where [ . ] denote the greatest integer function, find all possible coordinanates of A.
8) Let S be the square of unit area. Consider any quadrilateral which has one vertex on each side of S. if a, b, c and d denote the lengths of the sides of the quadrilateral, prove that : 2≤ a²+ b²+ c²+ d² ≤ 4.
9) If the points, {a³/(a-1), (a²-3)/(a-1)}, {b³/(b -1), (b²-3)/(b-1)} and {c³/(c -1), (c²-3)/(c -1)} are collinear for three distinct values a, b, c, and a≠ 1, b≠ 1 and c≠ 1, then show that
abc - (bc + ca+ ab) +3(a+ b + c)=0.
10) If A₁, A₂, A₃,.......Aₙ are n points in a plane whose coordinanates are (x₁, y₁), (x₂, y₂), (x₃,y₃), .......(xₙ, yₙ) respectively. A₁A₂ is bisected in the point G₁ ; G₁A₃ is divided at G₂ in the ratio 1:2; G₂A₄ is divided at G₃ in the ratio 1:3; G₃A₅ at G₄ in the ratio 1:4 and so on until all the points are exhausted. Show that the coordinanates of the final point so obtained are
(x₁+ x₂ +.....xₙ)/n and (y₁+ y₂+.....yₙ)/n
11) If by change of axes without change of origin, the expression ax²+ 2hxy + by² becomes a₁x₁²+ 2hx₁y₁ + b₁y₁² prove that
a) a+ b = a₁ + b₁.
b) ab - h²= a₁b₁ - h₁²
c) (a - b)²+ 4h² = (a₁ - b₁)² + 4h₁².
SLOPE OF A LINE
EXERCISE - 1
1) Verify that the points A(-1,3) B(0,5) And C(3,1) are the vertices of a right angled triangle.
2) A moving point A(x,y) remains always equidistant from A(-1,0) and B(0,-2). Express this fact by an algebraic relation in x and y. 2x - 4y -3=0
3) Verify by using the concept of slope, that the three points (5,7), (-3,1) and (-7,-2) are collinear .
4) A line L₁passes through the two points (3,4) and (2,1). If there is another line L₂ such that angle from L₁ to L₂ is 45°, show that the slope of L₂ is -2.
5) Find the interior angles of the triangle with vertices are (1,1), (5,2) and (4,3). 5/14, -5
6) Find the interior angles of a triangle with vertices (-3,-2), (2,5) and (4,2).
7) Find the slopes of the line joining the points:
a) (2,-5) and (5,4). 3
b) (-2,-7) and (-3,-1). -6
c) (2/3,5/2) at(-1/2, -1/3). 17/7
d) (-5/4, 4/3) and (3/4, 3/5). 11/30
8) Show that the three points in each of the following
a) (1,4), (3,-2) and (-3, 16)
b) (0 ,-2), (2,4) and (-1,-5)
are collinear
9) Find the angle between the straight lines y= mx + c and y₁= m₁x + c and hence deduce the condition of perpendicularity and parallelism of the lines.
10) Verufy the following statements :
a) The points (-1,1/2),(0,-5/2 and (5,5/2) are the vertices of a right triangle .
b) The four points (-4,0),(6,4),(5,0) and (0,-2) are the vertices of a trapezium.
11) The points (-3,3), (-1,-1) and (3,-3) are the vertices of an isosceles triangle.
12) The points (-5,-1), (0,-7),(18,9),(13, 15) are the vertices of a parallelogram.
13) The circle having the points (7,2) and (-3,2) as ends of a diameter also passes through (-1,6).
14) The perpendicular bisector of the line joining (-3,1) and (13, 3) passes through (7, -14).
15) The points (-4,0), (6,3) and (36,12) are collo.
16) Show that the quadrilateral with vertices (10,10),(-14, -2), (-10,- 10) and (4, -24) can be divided into two right triangles.
Prove that its area is 400 square units.
17) The point (x,y) is equidistant from (5,-2) and (-3,4 0). 4x - 3y -1=0
18) The point (x, y) lies on the circumference of a circle with the segment directed from (-3,6) to (2,5) as diameter. x²+ y²+ x - 11y +24 =0
19) A line L₁ passes through (-5,-3) and (2,6). Another line L₂ passes through (6,4) and (8,2). If θ be the angle from L₁ to L₂ , prove that tanθ = 8.
20) L₁ passes through (1,9) and (2,6); L₂ passes through (3,3) and (-1,5); to prove θ = π/4.
21) Find the interior angles of a triangle with vertices
a) (-10,-8),(11,6) and (37,23)
b) (7,12, (2,5) and (-5,-5). -1/105
22) A, B, C, D are the points (-2,0),(4,3),(1,-1) and (5,1); prove that AB|| CD.
23) A point P(x, y) lies on a line which passes through the point (2,3) and which is perpendicular to the line joining the points (-1,2) and (-5,4); show that 2x - y -1= 0.
THE STRAIGHT LINE:
ITS EQUATION IN DIFFERENT FORMS
EXERCISE - A
1) Find the equation of the line line passing through the point (2,-3) and perpendicular to the line joining the point (4,1) and (2,4). 3y - 2x +13=0
2) Find the equation of a straight line which passes through the (4,-8) and is inclined at an angle 135° to the positive x-axis. x+ y+ 4=0
3) Write down the equation of the straight lines:
a) inclined at an angle 60° to the positive x-axis and passing through (5,-2). y- √3 x + 2 + 5√3= 0
b) included at an angle whose trigonometric tangent is 3/4, and passing through (2,6). 4y - 3x = 18
c) of slope -4/5 and passing through the point (-2,4). 5y+ 4x -12=0
d) of slope m and passing through (0, a√(1+ m²)). y= mx + a √(1+ m²).
4) Find the equation of the line through (3,-6) perpendicular to the line joining (4,1) and (2,5). x - 2y = 15
5) Write the equation of a line through (3,-4) parallel to the line through (0,-5) and (4,-3). 2y - x +10= 10
6) Find the equation of the following lines:
a) passing through (0,0) and making an angle of 60° with the positive direction of axis of x. y= √3 x
b) Passing through (-4,0) and making an angle of 45° with the positive direction of axis of y. y - x - 4=0.
EXERCISE - B
1) Find the equation of a line passing through two points (3,4) and (8,-15). 5y+ 19x =77
2) What does the equation y= mx + c represent in each of the following case ?
a) when m= 0 and c is an arbitrary constant.
b) when c(≠0) is a fixed constant but m is an arbitrary constant.
c) when c=0 and m is an arbitrary constant
d) when m is a fixed constant (≠0) and c is also a fixed constant.
e) when both m and c are arbitrary constants.
3) Write down the equations of the lines joining the following points:
a) (0,0) and (5,-3). 3x - 5y= 0
b) (5,-3) and (5,2). x = 0
c) (-5,2) and (3,2). y= 0
d) (-4,1) and (3,-5). 6x +7y+ 17= 0
e) (a/m², 2a/m) and (a/4m², a/m). 3y = 4mx + 2a/m
f) (a,b) and (b, a). x + y= a+ b
g) (ct, c/t) and (2ct, c/2t). x + 2t²y= 3at
4) Find the equation of the sides of the triangle (produced (, formed with vertices (-5,6),(-1,-4) and (3,2). Derive the equations of the three medians. 5x + 2y +13= 0, 2y -3x +5= 0, x + 2y -7= 0, -x + 6y -9= 0, 7x + 6y -1 = 0
5) Show that the equation
x y 1
x₁ y₁ 1 = 0
x₂ y₂ 1
can be expressed as an equation of first degree in x and y.
EXERCISE - C
1) Find the equation to the line passing through (3,-4) and cutting of intercepts, equal but opposite signs , from the two axes. x - y =7
2) Find the equation to the line passing through (-5,4) and is such that the portion of it between two axes is divided by the point in the ratio 1:2. 5y - 2x = 30
3) a straight line passes through the point (4,3) and makes on the axes intercepts which are equal in magnitude and also in sign. Find the equation of the line and also the Intercepts on the axes. x + y=7
EXERCISE - D
1) Construct each of the following lines where
a) p= 5, α= 30°.
b) p= 4, α= 240°.
c) p= 5, α= 314°.
d) p= 6, α= 120°.
Write down the equation of each of these lines in the normal form.
EXERCISE - E
1. Reduce each of the equations to the normal form find and find a and p:
a) √3x + y=9. 30°, 9/2
b) x+ y+ 8= 0. 225°, 4√2
c) 4y -7=0. 90°, 7/4
d) x +5=0. 180°, 5
2) Determine the intercepts the following lines on each of the co-ordinate axes, wherever they exist and draw the lines:
a) 2x + 3y -12=0.
b) x - y + 1=0
c) 5x + 7y +13=0
d) 2x - 3y =0
e) 2x + 3=0
f) x= 0.
3) The diagonals of a square lie along the coordinate axes, and each has length 2 units. Find the equations of four sides (produced). x + y = ±1 and x - y= ±
4) find the equation to the diagonals of the rectangle the equation whose sides are x= a, x= A' , y= b, y= b'. y(a'- a)- x(b' - b)= a'b - ab'; y(a'- a)+ x(b' - b)= a'b - ab';
5) find the equation to the straight line which go through the origin and trisect the portion of the 3x+ y=12 which is intercepted by the axes. y= 6x, 2y= 3x,
6) find the equation to the line which bisects the distance between the points (a,b) and (A', b') and bisects the distance between the points (-a, b) and (A', -b'). 2ay - 2b'x = ab - a'b
EXERCISE - F
1) In the triangle with vertices (0,6),(-2,-2),(4,2), find
a) the equations of the medians and their point of intersection. y+ 6x -6=0, 2y -3x -2=0, y=2. (2/3,2)
b) the equations of their altitude and their point of intersection. 2y+3x -12=0, 4y+ x -12=0,
c) the equation of the perpendicular bisectors of the side. 2y+ 3x -3=0
Verify that the 3 points of intersection so found lie on a straight line.
2) In the triangle with vertices (2,0),(3, 2),(4,-3) find
a) the equations of the medians and their point of intersection
b) the equations of their altitude and their point of intersection
c) the equation of the perpendicular bisectors of the sides and their point of intersection .
Verify that the 3 points so found lie on straight line.
EXERCISE - G
1) Find the equation of a straight line, which passes through the point (5,-6) and is
a) parallel to the line 8x + 7y +5= 0. 8x + 7y +2= 0
b) perpendicular to the line 8x + 7y +5= 0. 7x -8y -83 = 0
2) Find the angle between the lines
a) x - y√3= 5 and √3x + y= 7. 90°
b) x - 4y -3= 0 and 6x -y = 11. tan⁻¹(23/10)
c) y= 3x +7 and 3y - x = 8. tan⁻¹(4/3)
d) y= (2- √3)x + 5 and y= (2+ √3)x -7. 60°
3) Find the trigonometrical tangent of the angle between the lines whose intercepts on the axes are respectively a, - b and b, -a. tan⁻¹{(a²- b²)/2ab}
4) Prove that 4 points (2,1), (0,2),(2,3) and (4,0) form a parallelograms and that the angle between its diagonals is. tan⁻¹(2).
5) Find the equation to the straight line:
a) passing through (4,-5) and parallel to the line passing through 3x + 4y +5=0. 3x + 4y +8=0
b) passing through (4,-5) and perpendicular to the line 3x + 4y +5=0. 4x - 3y = 31
c) passing through (2,-3) and perpendicular the line joining the points (5,7) and (-6,3). 4y + 11x = 10
6) Find the equation to the straight line drawn at right angles to the straight line x/a - y/b =1 (a, b > 0) through the point where it meet the axis of x. ax + by = a²
7) Find the equation to the straight line which bisects , and is perpendicular to, the straight line joining the point (a,b) and (c,d). 2x(a- c)+ 2y(b - d)= a²- c²+ b²- d².
8) Find the equation of a line passing through (x₁, y₁) and Perpendicular to the lines respectively
a) yy₁ = 2a(x + x₁). 2ay + xy₁ = 2ay+ x₁y₁
b) xx₁ + yy₁ = a². x₁y - xy₁= 0
c) xx₁/a² + yy₁/b² = 1. a²xy₁ - b²x₁y = (a²- b²)x₁y₁
d) x₁y + xy₁ = a². xx₁ - yy₁ = x₁²- y₁²
9) Prove that the equation of a line which passes through (a cos³ θ, a sin³ θ) and is perpendicular to the line x secθ + y cosecθ = a is x cosθ - y sinθ = a cos2θ.
10) Find the equation of the line perpendicular to 2x - y -4=0 and cutting from the first quadrant a triangle whose area is 16 square units. x + 2y -8=0
11) A line is parallel to the line 3x + 2y -6=0, and forms a triangle in the first quadrant with the lines.
x - 2y=0, and 2x -y= 0, whose area is 21 square units. Find the equation of the line. 3x + 2y -28= 0
EXERCISE - H
1) Find the equations to the straight lines which pass through (2,3) and make acute angle 45° with the line 3x - y +5=0. 2y - x -4=0, y+ 2x -7=0
2) Prove that the equation to the straight lines passes through (3,-2) and inclined at 60° to the line √3x + y =1 are y+2= 0 and y - √3 x + 2 + 3√3=0.
3) Prove that the equations to the lines which pass through a given point (x', y') and makes acute angle α with the given line y= mx + c, are
y- y'= (m± tanα)(x - X')/(1± tanα).
4) Find angle measured from the straight line 3x= 4y + 7 to the line 5y = 12x +6 and also the equations to the straight lines which pass through the point (4,5) and make equal angles with the two given lines. tan⁻¹(33/56); 9x - 7y = 1, 7x + 9y = 73.
EXERCISE - I
1) Prove that the origin and (2,3) are on the opposite sides of the line 7x - 24y +8=0.
2) Show that the points P:(3,2), Q(3,-1) lie within adjacent angles formed by the lines x - 2y +2= 0 and x + y +1=0.
3) Show that the four points (0,0),(-1,1),(-7,4) and (9,6) are in the four different compartments made by the two straight lines 2x - 3y +1=0, and 3x - 5y +2=0.
4) Show that the points (3,2) and (7,3) lie on the opposite sides of the line: 2x - 5y +3=0.
5) Show that the point (-2,6) lies on the negative side of the line 3x + 2y -7= 0.
6) Show that the origin is within the triangle the equations of whose sides are x - 7y +25= 0, 5x + 3y +11= 0 and 3x - 2y -1= 0.
Perpendicular distance of a point from a line
EXERCISE - J
1) Find the distance of the given point from the given line:
a) (5,2) ; 3x - 4y +6=0. 13/5
b) (6,-1) ; 3x - y +1=0. 2√10
c) (3,4) ; 2x +5 =0. 11/2
d) (-2,-5) ; y =0. 5
e) origin; 3x +2y -6=0. 6√13/13
2) Find the lengths of the altitude of the triangle with vertices in the points (2,0),(3,5),(-1,2). 17√26/26,17/5,5√13/13
EXERCISE - K
1) Find the equation to the bisectors of the angles between the straight line 3x -4y -2=0 and 5x +12y +6=0. 16x +2y+1=0; 8y- x +4=0
2) Find the centre of the circle inscribed in the triangle whose angular points A, B, C are rspectively (1,2,(25,8),(9,21). (23/2,11)
3) Find the equation to the bisectors of the angles between the lines
a) 13x -9y =10 and 11x+ 2y =6. 2x -6y+5=0;9x +3y=20.
b) 2x +y =10 and x+ 3y =2. 21x +7y+3=0; x -3y=7.
c) y =3x and x+ y =1. (3±√5)x -(1±√5)y= ±√5
d) 3x -4y =-7 and 12x-5y =8. 99x -77y+51=0; 21x +27y=131.
e) y- b = 2m(x - a)/(1- m²) and y- b= 2m'(x -a)/(1- m'²).
4) Find the equations to the bisectors of the internal angles of the triangle the equations of the sides are 3x +5y =15 , x+ y =4 and 2x +y+5=6.
5) If the equation of sides 7x -y +11=0; x+ y =15; 7x +17y+ 65=0. (5,1)
PARAMETRIC EQUATION OF A STRAIGHT LINE
EXERCISE - L
1) A line of slope 3/4 passing through a point P(-2,-5). Find the coordinates of any point Q(x,y) on the line, given PQ= 10 units. (6,1) Or (-10,-11)
2) Find the direction in which a straight line must be drawn through the point (1,2) so that its point of intersection with the line x+ y =4 may be at a distance √6/3 from this point. 15° or 75°
3) A is the point (-5,-3) and B lies on the line x - 3y -1=0. AB is inclined to the x-axis at an acute angle whose tangent is 5/12. Find the length of AB. 13 units
4) A line of love 5/12 passes through A(3,-7/2), Find the coordinates of a point B on the line where AB = 13/2. (9,-1) Or (-3,-6)
5) Find the points on the following lines at which the x and the y co-ordinates are equal, and hence write each equation in the form (x - x₁)/cosθ = (y - y₁)/sinθ = r
a) 4x - 3y -1=0. (x-1)/3/5 - (y-1)/4/5= r
b) 5x +12y +17=0. (x+1)/12/13 = (y+1)/-5/13= r
c) x - √3y =0. x/√3/2 = y/1/2= r
EXERCISE - M
1) Find the equation to the straight line passing through the point of intersection of the lines x + 2y + 3 = 0 and 3x + 4y + 7=0 and perpendicular to the straight line x - y + 8=0. x+ y+2=0.
2) Show that the lines x - 4y+2=0, 4x- y+3 =0, x+ 2y=0 meets at a point.
MISCELLANEOUS - A
1) Find the equation of a straight line passing through (1, 2) and inclined an angle 135° with the positive direction of the x-axis. x+ y -3=0.
2) Find the equation of the straight line passing through the points of intersection of the curves (y+1)²+ 4x =0 and x²- 4(y+1)= 0. x+ y+1=0.
3) Find the perpendicular distance from the origin of the straight line 3x+ 4y= 5√2; also find the angle that this perpendicular makes with the positive direction of the x-axis . tan⁻¹(4/3)
4) The equation of a straight line is 5x+ 2y =0; this equation can be expressed in the intercept from --- discuss the validity or otherwise of the statement.
5) Obtain the equation to the lines which pass through the origin and trisect the segment of the line 3x+ y -12=0 intercepted between the axes. y= 6x
6) Find the area of the triangle formed by the straight line 2x+ 3y -12=0 with the x-axis of coordinates. 12
7) if a + b + c = 0, show that the three lines ax+ by + c =0, bx + c y + a=0, and cx + ay + b= 0 are concurrent.
8) Find the equation of the straight line which passes through the point of intersection of the line 2x+ 3y -5=0 and 3x+ 5y -7=0 and makes equal positive intercs upon the coordinate axes. x+ y=3
9) Find the distance between the parallel lines y= mx + c₁ and y= mx + c₂.
10) Find the distance measured along the line 4x -3y+2=0 from the point (1,2) to the line x- 2y-2=0. 5
11) Find the locus of the middle point of the portion of the line-segment made by the straight line x cosα + y sinα =4 and the axes of coordinates. 4(x²+ y²)= x²y²
12) Find the distance of the point (3,5) from the line 2x+ 3y-14=0 measured parallel to the line x -2y-1=0. √5
13) A variable line drawn through the point of intersection of the lines x/a + y/b =1 and x/a + y/a = 1 meets the coordinate axes at P and Q. Show the locus of the midpoint of PQ is the curve 2xy(a+ b)= ab(x + y).
14) Show that the diagonals of the parallelogram formed by the lines √3x+ y=0, √3x+ y -1=0, √3y+ x=0, √3 y+ x=1 are at right angles.
MISCELLANEOUS - B
1) What does the equation y= mx + c represent in each of the following cases:
a) When m= 0 and c is an arbitrary constant;
b) When c≠ 0 is a fixed constant but m is an arbitrary constant.
c) When c= 0 and m is an arbitrary constant.
d) When m≠ 0 is a fixed constant and c is also a fixed constant .
e) When both m and c are arbitrary constants.
2) Find the angle which the straight line perpendicular to the line √3x + y=1 makes with the positive direction of the x-ai.
3) A straight line makes Intercepts h and k upon the co-ordinate axes; find its equation. What is the length of the perpendicular from the origin upon the line ?
4) Is it possible to express the equation to a line parallel to a coordinate axis in terms of intercept form x/a + y/b =1?
5) State with reasons whether the equation of
a) straight lines through the origin.
b) straight line parallel to the x-axis can be expressed in the form of x/a + y/b= 1
6) The equation of a straight line is 5x+ 2y=0, this equation can be expressed in the intercept from -- discuss the validity or otherwise of the statement .
7) Prove that a linear equation of the form ax+ by+ c =0 always represents a straight line on a plane.
8) What does ax+ by+c=0 represent, when
a) a≠0, b≠ 0, c= 0
b) a= 0, b≠ 0, c≠ ó
c) a= 0, b≠ 0, c= ó
d) a ≠0, b= 0, c≠ ó
e) a≠ 0, b= 0, c= ó
f) a≠ 0, b≠ 0, c≠ ó
9) Find the co-ordinates of the point of intersection of the lines ax+ by+ c =0 and a'x+ b'y+ c' =0.
10) Represent any line passing through the point of intersection 2 lines ax+ by+ c =0 and a'x+ b'y+ c' =0 in the form of an equation.
11) The perpendicular distance of a straight line from the origin is P and the perpendicular make an angle α with the positive direction of the x-axis. Find the equation of the straight line.
12) Find the perpendicular distance of a point from a line.
13) Reducing the straight line 3x+ 4y+ 15 =0 to its normal form, find the perpendicular distance of the line from the origin.
14) Find the angle between the straight line y= m₁x + c₁ and y= mx₂ + c₂ and hence deduce the condition of perpendicularity and parallelism of the lines. Do the condition change when lines become y= m₁ and y= m₂x ? find also the conditions when lines are ax+ by+ c =0 and a'x+ b'y+ c' =0.
15) Find the equation of the bisectors of the angles between two lines ax+ by+ c =0 and a'x+ b'y+ c' =0.
16) Find the condition that the three lines a₁x + b₁y + c₁=0, a₂x + b₂y + c₂ =0 and a₃x + b₃y + c₃ =0be concurrent.
1) Two equal arcs of two circles subtends angle of 60° and 75° at the centre. Find the ratio of the radii of the two circles.
3) If 7 cosθ + 5sinθ= 5, find 5cosθ - 7sinθ.
4) If secθ+ tanθ= x, show that sinθ = (x²-1)/(x²+1).
5) If sinθ + cosecθ= 2 show that sinⁿθ + cosecⁿθ= .
6) If tan⁴θ+ tan²θ= 1, show that cos⁴θ+ cos²θ= 1.
7) If If cis⁴θ+ cos²θ= 1, show that tan⁴θ+ tan²θ= 1.
8) If sinα, cosα, tanα are in GP show that cot⁶α- cot²α= 1.
9) If (secx -1)(secy -1) secx -1)= (secx +1)(secy +1)(secx+1) show the value of each side ± tanx tan y tan z.
10) If tanθ+ sinθ= m and tanθ - sinθ= n show that m²- n²= 4√(mn).
11) If x=cosecα - sinα and y= secα - cosα, show that x²y²(x²+ y²+3)= 1.
12) If cosecα + cosec β + cosecγ = 0, then show (sinα sinβ sinγ)²= sin²α +sin² β + sin²γ.
13) Find the least value of 9 tan²θ + 4 co5²θ.
14) If θ lies in the 2nd quadrant and tanθ = - 5/2 find the value of 2 cosθ/(1- sinθ).
15) If sinθ = 8/15 and sinθ is negative find {sin(θ)+ cos(-θ)}{sec(-θ)+ tan(-θ)}.
16) If α= π/19, show that (sin23 - sin3α)/(sin16α + sin4α)= +1.
17) Evaluate {cot570°+ sin(-330°)/tan(-210°) + cosecx(-750°).
18) If n be any integer; show sin{nπ+(-1)ⁿ π/4}= 1/√2.
19) If α and β are positive acute angls and cos α = 1/√10 and sinβ= 1/√2 find α - β.
20) If x,y are positive acute angles and cosecx =√5, secy = √10/3 find cosecx(x -y).
