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

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