Question

Ordinates of three points $A,B,C$ on the parabola $y^2=4ax$ are in G.P. Tangents at $A$ and $C$ intersect on

• A. line through $B$ parallel to $x-$axis
• B. line through $B$ parallel to $y-$axis
• C. line through $B$ and vertex of parabola
• D. line through $B$ and focus of parabola

Answer 1

The correct option is B line through $B$ parallel to $y-$axis

Let the three points $A,B$ and $C$ on the parabola are $A(x_1,y_1),B(x_2,y_2)$ and $C(x_3,y_3)$ respectively.

Also ${y_2}^2=y_1{y}_2$ (given)

If $T$ be the point of intersection of tangents at $y_1$ and $y_2$ then $T$ is

$T(\frac{y_1{y}_3}{4a},\frac{y_1+y_3}{2})=T(\frac{{y_2}^2}{4a},\frac{y_1+y_3}{2})=T(x_2,\frac{y_1+y_3}{2})(\because {y_2}^2=4a{x}_2)$

This point lies on $x=x_2$ which is a line though $B(x_2,y_2)$ parallel to y-axis.