Question

The condition that the line $\frac{x}{p}+\frac{y}{q}=1$ be a normal to the parabola $y^{2}$ = 4ax is]

• A. $p^{3}=2ap^{2}+aq^{2}$
• B. $p^{3}=2aq^{2}+ap^{2}$
• C. $q^{3}=2ap^{2}+aq^{2}$
• D. None of these

Answer 1

Answer: (A)

The line $\frac{x}{p}+\frac{y}{q}=1$ be a normal to the parabola $y^{2}=4 a x$
if, for some value of $m$, it is identical with
$y=m x-2 a m-a m^{3} \text { i.e. } m x-y=\left(2 a m+ a m^{3}\right)$
Comparing coefficients, we get
$\frac{m}{1 / p}=\frac{-1}{1 / q}=\frac{2 a m +a m^{3}}{1} \rightarrow m p=-q$
$\therefore m=-q / p \text { and } m p=m\left(2 a +a m^{2}\right)$
or $p=2 a+\frac{a q^{2}}{p^{2}} \text { or } p^{3}=2 a p^{2}+a q^{2}$