2) Write down the equation of the tangent of the parabola y²=4x at (1,2). x - y +1= 0.
3) Find the equation the normal to the curve xy= c² at (ct, c/t). t³x -yt= c(t⁴-1)
4) Write down the equation of the normal to the parabola y²=25x at point (1,-5). 2x- 5y = 27
5) Write down the length of the tangent drawn from an external point (a, b) to the circle x²+ y²+ 2x =0. a²+b²+ 2a units
6) Write down the value of slope of the tangent to the parabola y²= 8(x -6) at the point (8,-4). -1
7) Determine the points on the curve y= x + 1/x, where are the tangent is parallel to the x-axis. (1,2) and (-1,-2)
8) Find the gradient of the tangent to the parabola y²= 4x at the point (1,2). 1
9) Show that the equation of the normal to the hyperbola x = a sec k, y = b tan k at the point (a sec k, b tan k) is ax cos k + by cot k = a²+ b².
10) Find the condition that the straight line lx + my = n touches is the ellipse x²/a² + y²/b² =1. a²l²+ b²m²= n²
11) Write down the equation of the tangent and normal of the parabola y²= 4ax at (0,0). x=0, y= 0
12) Obtain the equation of the normal to the hyperbola x²/b² - y²/b² =1 at (a sec k, b tan k). ax cos k+ by cot k = a²+ b²
13) If the straight line lx + my =1 is a normal to the parabola y²=4ax, then show that al³+ 2alm²= m².
14) Find the equation of the tangent to the hyperbola x²/a² - y²/b²= 1 at (a sec k, b tan k). x² - y²= 1
15) Find the equation of the tangent and normal to the ellipse 4x² +9y²= 72 at the point (3,2). 2x+ 3y = 6, 3x- 2y = 5
16) Find the equation of the tangent to the hyperabola y²= 4x+ 5, which is parallel to the straight line y= 2x +7. y= 2x+3
17) Find the equation of the normal to the curve y= x²- x at the point (3,6). Show that this normal touches the parabola x²+ 660y=0. x+ 5y = 33.
18) If the straight line lx + my = n is normal to the hyperbolas x²/a² - y²/b² = 1, then show that a²/l² - b²/m²= (a²+ b²)²/n²
19) If the straight line x cos k + y sin k = p touches the ellipse x²/a² + y²/b² = 1, then prove that a² cos²k + b² sin²k = p²
20) Prove that the straight line x+ y= 2+ √2 touches the circle x²+ y² - 2x - 2y +1= 0. Find the point of contact. (1+ 1/•2, 1+ 1/√2)
21) If the straight line lx+ my + n =0 touches the parabola y²= 4ax then prove that am² = nl.
22) Find the equation of the tangent and the normal to the curve y= x³- 3x at the point (2,2). 9x-y-16=0, x+ 9y -20= 0.
23) Show that the condition that the two curves ax²+ by² = 1 and a'x²+ b' y² =1 (ab'- a'b ≠ 0) intersects orthogonally is 1/a - 1/b = 1/a' - 1/b'.
24) Find the equation of the tangent and normal of the curve y(x -2)(x -3) - x +7 = 0 at its point of intersection with the x-axis. x- 20y-7=0; 20x + y=140.
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