Paragraph: 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 Q R and

Correct answer: D. t ^ 2 - 1 t Slope (QR) = slope (PK) 2at - 0 a t ^ 2 - 2a = - (2a)/(t) - 2ar a t ^ 2 - a r ^ 2 ⇒ t t ^ 2 - 2 = - ( (1)/(t) + r 1 t ^ 2 - r ^ 2 ) ⇒ r = t ^ 2 - 1 t

JEE Advanced 2014Paper 2

Exam: JEE Advanced
Year: 2014
Paper: Paper 2
Subject: Mathematics
Chapter: Parabola

JEE Advanced 2014 · Paper 2 · Mathematics · Parabola

Paragraph: Let a, r, s, t be nonzero real numbers. Let P(a t^ 2 , 2...

Paragraph: 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)

.
Question: The value of

r

Options

A $- \dfrac{1}{t}$
B $\dfrac{{t}^{2} + 1}{t}$
C $\dfrac{1}{t}$
D $\dfrac{{t}^{2} - 1}{t}$

Correct answer

D. $\dfrac{{t}^{2} - 1}{t}$

Step-by-step solution

Slope (QR) = slope (PK)

\dfrac{2at - 0}{a{t}^{2} - 2a} = \dfrac{- \dfrac{2a}{t} - 2ar}{\dfrac{a}{{t}^{2}} - a{r}^{2}}

⇒ \dfrac{t}{{t}^{2} - 2} = - \left(\dfrac{\dfrac{1}{t} + r}{\dfrac{1}{{t}^{2}} - {r}^{2}}\right) ⇒ r = \dfrac{{t}^{2} - 1}{t}

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