Question

let $P$ be the point $(1,0)$ and $Q$ a point on the locus $y^2=8x$. The locus of mid point of $PQ$ is

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

Answer 1

The correct option is D $y^2-4x+2=0$

Let $M(x^{\prime },y^{\prime })$ be point of locus, so mid point of $PQ,$ $\Rightarrow \frac{x^{\prime }+1}{2}=h,\frac{y^{\prime }+0}{2}=k$ $\Rightarrow x^{\prime }=2h-1,y^{\prime }=2k$

Now $(x^{\prime },y^{\prime })$ lies on $y^2=8x$

$\Rightarrow (2k{)}^2=8(2h-1)$
$\Rightarrow y^2=2(2x-1)$

$\Rightarrow y^2-4x+2=0$