IIT JEE , AIEEE Forum - Mathematics , Analytical Geometry

sachin_gupta1991

Q. The condition for the two tangents drawn from a point to the parabola y²=4ax,

to become normals to the parabola x²=4by is that a²>8b^2.prove this.

feynmann

Eqn of normal at point ' t' of the parabola x^2 = 4by is

x + ty = 2bt + bt^3 .................( 1 )

Eqn of tangent at ( x1 , y1 ) of the parabola y^2 = 4ax is

yy1 = 2a ( x + x1 ) ................... ( 2 )

comparing ( 2 ) with ( 1 ) , we get

2a = - y1 / t = 2ax1 / ( 2bt + bt^3 ) ................ ( 3 )

Now( x1 , y1 ) satisfies y1^2 = 4ax1

so , from (3) we get

bt^2 - at + 2b = 0

we will get two normals provided discriminant of the above eqn > 0

giving a^2 > 8b^2 ( proved )