Question

If the normals at two points $P$ and $Q$ of a parabola $y^2=4ax$ intersect at $R$ on the curve, then the product of ordinates of $P$ and $Q$ is

• A. $4a^2$
• B. $2a^2$
• C. $-4a^2$
• D. $8a^2$

Answer 1

The correct option is D $8a^2$

If the normal at two points $P\left(t_1\right)$ and $Q\left(t_2\right)$ of a parabola $y^2=4ax$ intersect at $R$ on the curve, then
$\Rightarrow t_3=-t_1-\frac{2}{t_1}$
Also, $t_3=-t_2-\frac{2}{t_2}$
​​​​​​​$\Rightarrow -t_1-\frac{2}{t_1}=-t_2-\frac{2}{t_2}$
​​​​​​​​​​​​​​$\Rightarrow t_2-t_1=2(\frac{1}{t_1}-\frac{1}{t_2})$
$t_1{t}_2=2$
$\therefore$ Product of ordinates $=2a{t}_1\times 2a{t}_2=8a^2$