Question

If the normal at $(ap^2, 2ap)$ on the parabola $y^2 = 4ax,$ meets the parabola again at $(aq^2, 2aq)$, then

• A. $p^2 + pq + 2 = 0$
• B. $p^2 - pq + 2 = 0$
• C. $q^2 + pq + 2 = 0$
• D. $p^2 + pq + 1 = 0$

Answer 1

Answer: (A)

$p^2 + pq + 2 = 0$

Answer 2

Since the normal at $(ap^2, 2ap)$ on $y^2 = 4ax$ meets the curve again at $(aq^2, 2aq)$, therefore $px + y = 2ap + ap^3$ passes through $(aq^2,2aq)$
$\Rightarrow \; paq^2 + 2aq = 2ap + ap^3$ $\Rightarrow \; p(q^2-p^2) = 2(p - q)$ $\Rightarrow \; p (q + p) = -2$ $\Rightarrow \; p^2 + pq + 2 = 0$