Mathematics | VITEEE 2017 | Discussion Page

The condition that the line $\frac{x}{p}+\frac{y}{q}$ = 1 be a normal to the parabola y²= 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

Option A

The line $\frac{x}{p}+\frac{y}{q}$ = 1 be a normal to the parabola y²= 4ax if, for some value of m, it is identical with

y = mx-2am - am³ i.e. mx-y = (2am+ am³)

Comparing coefficients, we get

$\frac{m}{1/p}=\frac{-1}{1/q}=\frac{2am +am^{3}}{1}$ → mp = -q

.'. m= -q/p and mp = m(2a+am²)

or p = $2a+\frac{aq^{2}}{p^{2}}$ or $p^{3} = 2ap^{2}+aq^{2}$

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