The chord `A B` of the parabola `y^2=4a x` cuts the axis of the parabola at `Cdot` If `A-=(a t_1 ^2,2a t_1),B-=(a t_2 ^2,2a t_2)` , and `A C : A B 1:3,` then prove that `t_2+2t_1=0` .

If the normal to the parabola y^2=4a x at point t_1 cuts the parabola again at point t_2 , then prove that t_2 2geq8.

If t_1a n dt_2 are the ends of a focal chord of the parabola y^2=4a x , then prove that the roots of the equation t_1x^2+a x+t_2=0 are real.

If the normals at points t_1a n dt_2 meet on the parabola, then t_1t_2=1 (b) t_2=-t_1-2/(t_1) t_1t_2=2 (d) none of these

Find the equation of the tangent at t =2 to the parabola y^(2) = 8x .

if the normal at the point t_(1) on the parabola y^(2) = 4ax meets the parabola again in the point t_(2) then prove that t_(2) = - ( t_(1) + 2/t_(1))

If the normals at P(t_(1))andQ(t_(2)) on the parabola meet on the same parabola, then

Statement 1: The line a x+b y+c=0 is a normal to the parabola y^2=4a xdot Then the equation of the tangent at the foot of this normal is y=(b/a)x+((a^2)/b)dot Statement 2: The equation of normal at any point P(a t^2,2a t) to the parabola y^2 = 4ax is y=-t x+2a t+a t^3

Find the derivatives of the following : x = (1-t^2)/(1+t^2), y = (2t)/(1+t^2)

Find the locus of the point (t^2-t+1,t^2+t+1),t in Rdot