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

Answer: (B)

line through $B$ parallel to $y-$axis

Let the coordinates of $A,B,C$ be $(a{t}_i^2,2a{t}_i)$ where $i=1,2,3$

$A,B,C$ have orrdinates in GP so that

$t_1{t}_2{t}_3$ are also in GP i.e. $t_1{t}_3=t_2^2$

Equations of tangents at $A$ and $C$ are

$t_{1}y=x+a{t}_1^2$ and $t_{3}y=x+a{t}_3^2$ which intersect at point

$\frac{x+a{t}_1^2}{t_1}=\frac{x+a{t}_3^2}{t_3}\Rightarrow x=a{t}_1{t}_3=a{t}_2^2$

This is a line through $B$ parallel to y axis.