Let $a, r, s, t$ be nonzero real numbers. Let $P\left(a t^{2}, 2 a t\right), Q, R\left(a r^{2}, 2 a r\right)$ and $S\left(a s^{2}, 2 a s\right)$ be distinct points on the parabola $y^{2}=4 a x$. Suppose that $P Q$ is the focal chord and lines $Q R$ and $P K$ are parallel, where $K$ is the point $(2 a, 0)$. The value of $r$ is

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)

.

The value of

r

Answer: (D)

Since

P Q

is a focal chord and

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

, the other end

Q

of the focal chord is

Q \equiv (\frac{a}{t ^{2}}, - \frac{2 a}{t})

.

The slope of

P K

where

K = (2 a, 0)

is:

m_{P K} = \frac{2 a t - 0}{a t ^{2} - 2 a} = \frac{2 a t}{a ( t ^{2} - 2 )} = \frac{2 t}{t ^{2} - 2}

The slope of

QR

is:

m_{QR} = \frac{2 a r - ( - \frac{2 a}{t} )}{a r ^{2} - \frac{a}{t ^{2}}} = \frac{2 a ( r + \frac{1}{t} )}{a ( r ^{2} - \frac{1}{t ^{2}} )} = \frac{2 ( r + \frac{1}{t} )}{r ^{2} - \frac{1}{t ^{2}}}

Since

QR ∥ P K

, their slopes are equal:

\frac{2 ( r + \frac{1}{t} )}{r ^{2} - \frac{1}{t ^{2}}} = \frac{2 t}{t ^{2} - 2}

Simplifying gives

r = \frac{t ^{2} - 1}{t}

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