Question

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

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

Answer 1

Answer: (A)

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