Question

Let $P$ be the midpoint of a chord joining the vertex of the parabola $y^{2}=8x$ to another point on it. Then the locus of $P$ is

• A. $y^{2}=2x$
• B. $y^{2}=4x$
• C. $\frac{x^{2}}{4}+y^{2} =1$
• D. $x^{2}+\frac{y^{2}}{4}=1$

Answer 1

Answer: (B)

Let the chord be $M N$.
Let other end of chord joining from vertex to it lying on
$y^{2}=8 x$ be $N\left(2 t^{2}, 4 t\right)$

$\therefore $ Mid-point of $M N=\left(t^{2}, 2 t\right)$
$\therefore x=t^{2}, y=2 t \Rightarrow x=\left(\frac{y}{2}\right)^{2}$
$\Rightarrow y^{2}=4 x$
is the required locus of $P$.