Question

If $P,Q,R$ are three points on a parabola $y^2=4ax$ whose ordinates are in geometric progression, then tangents at $P$ and $R$ meet on the

• A. tangent at vertex of parabola
• B. abscissa of $Q$
• C. directrix of parabola
• D. ordinate of $Q$

Answer 1

The correct option is B abscissa of $Q$

Let the co-ordinates of $P,Q,R$ be $(a{t_i}^2,2a{t}_i);i=1,2,3$ having co-ordinates in $G.P.$
So that $t_1,t_2,t_3$ are also in $G.P.$ i.e., $t_1{t}_3={t_2}^2$

Equations of the tangent at $P$ and $R$ are

$t_{1}y=x+a{t_1}^2$ and $t_{3}y=x+a{t_3}^2$
which intersects at the point

$\frac{x+a{t_1}^2}{t_1}=\frac{x+a{t_3}^2}{t_3}$
$\Rightarrow x=a{t}_1{t}_3=a{t_2}^2$
which is a line through $Q$ parallel to $Y-$axis.

Hence, option B.