JEE Advanced · 2014 · 25 May 2014 · Paper 2 · Parabola

Let

a, r, s, t

be nonzero real numbers. Let

P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)

and

S (a s^{2}, 2 a s)

be distinct points on the parabola

y^{2} = 4 a x

. Suppose that

P Q

is the focal chord and lines

QR

and

P K

are parallel, where

K

is the point

(2 a, 0)

.

If

s t = 1

, then the tangent at

P

and the normal at

S

to the parabola meet at a point whose ordinate is

Answer: (B)

Solution

Given

s t = 1

, so

s = \frac{1}{t}

.

The equation of the tangent at

P (a t^{2}, 2 a t)

is:

t y = x + a t^{2}

The equation of the normal at

S (a s^{2}, 2 a s) = (\frac{a}{t ^{2}}, \frac{2 a}{t})

is:

\frac{x}{t} + y = \frac{2 a}{t} + \frac{a}{t ^{3}}

Solving these two equations simultaneously:
From the tangent equation:

x = t y - a t^{2}

.

Substituting into the normal equation:

\frac{t y - a t ^{2}}{t} + y = \frac{2 a}{t} + \frac{a}{t ^{3}}

y - \frac{a t ^{2}}{t} + y = \frac{2 a}{t} + \frac{a}{t ^{3}}

2 y = \frac{2 a}{t} + \frac{a}{t ^{3}} + a t

y = \frac{a ( t ^{2} + 1 ) ^{2}}{2 t ^{3}}

JEE Advanced · 2014 · 25 May 2014 · Paper 2 · Single correct option