Question

Let $P$ be the mid-point 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

The correct option is B $y^2=4x$

Let the chord be $MN$.
Let other end of chord joining from vertex to it lying on $y^2=8x$ be $N(2t^2,4t)$
$\therefore$ Mid point of $MN=(t^2,2t)$
$\therefore x=t^2,y=2t\Rightarrow x={\left(\frac{y}{2}\right)}^2$
$\Rightarrow y^2=4x$
is the required locus of $P$.