Question

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

• A. the line through $Q$ parallel to $x-$axis
• B. the line through $Q$ parallel to $y-$axis
• C. the line joining $Q$ to the vertex
• D. the line joining $Q$ to the focus.

Answer 1

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

Let the coordinates of $P,Q,R$ be $(a{t}_i^2,2a{t}_i),i=1,2,3$ having 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 tangents at $P$ and $R$ are $t_{1}y=x+a{t}_1^2$ and $t_{3}y=x+a{t}_3^2$, which intersect 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.