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

Answer 1

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

Let $P(1,0)$ and $Q(h,k)$ be a point on the locus

$k^2=8h$

Let $(\alpha ,\beta )$ be the midpoint of $PQ$

$\Rightarrow \alpha =\frac{h+1}{2},\beta =\frac{k+0}{2}$

$\Rightarrow 2\alpha -1=h,2\beta =k$

$\Rightarrow {\left(2\beta \right)}^2=8(2\alpha -1)$

$\Rightarrow {\beta }^2=4\alpha -2$

$\Rightarrow y^2-4x+2=0$ is the locus of the midpoint $PQ$