Q. If the normal at $(ap^{2},\ 2ap)$ on the parabola $y^{2} = 4ax$, meets the parabola again at $\left( aq^{2},\ 2aq \right),$ then
A) $p^{2} + pq + 2 = 0$
B) $p^{2} - pq + 2 = 0$
C) $q^{2} + pq + 2 = 0$
D) $p^{2} + pq + 1 = 0$
Explanation
(a)
We know that the normal drawn at a point $P\left( {at}_{1}^{2},2{at}_{1} \right)$ to the parabola $y^{2} = 4ax$ meets again the parabola at $\mathcal{Q}\left( {at}_{2}^{2},2{at}_{2} \right)$, then
$t_{2} = - t_{1} - \frac{2}{t_{1}}$
Here, $t_{1} = P\ and\ t_{2} = Q$
$\therefore q = - p - \frac{2}{p}$
$\Rightarrow \ p^{2} + pq + 2 = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.