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JEE Main 2005

Exam: JEE Main
Year: 2005
Subject: Mathematics
Chapter: Parabola

JEE Main 2005 · Mathematics · Parabola

Let P be the point (1,0) and Q a point on the locus y^2=8 x . The...

Let

P

be the point

(1,0)

and

Q

a point on the locus

y^2=8 x

. The locus of mid point of

P Q

is

Options

A $y^2-4 x+2=0$
B $y^2+4 x+2=0$
C $x^2+4 y+2=0$
D $x^2-4 y+2=0$

Correct answer

A. $y^2-4 x+2=0$

Step-by-step solution

\mathrm{P}=(1,0)Q=(h, k)

such that

k^2=8 h

Let

(\alpha, \beta)

be the midpoint of PQ

\begin{aligned}& \alpha=\frac{\mathrm{h}+1}{2}, \quad \beta=\frac{\mathrm{k}+0}{2}\\ & 2 \alpha-1=\mathrm{h}, \quad2 \beta=\mathrm{k} .\\ & (2 \beta)^2=8(2 \alpha-1) \Rightarrow \beta^2=4 \alpha-2 \\ & \Rightarrow \mathrm{y}^2-4 \mathrm{x}+2=0 . \end{aligned}

$

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