Question

If $Q$ be a point on the parabola $y^2=8x$ and $P(-2,0)$ be a point in the $xy$ plane. If the locus of the mid point of $PQ$ is a parabola, then its focus is

• A. $(1,1)$
• B. $(0,0)$
• C. $(-1,0)$
• D. $(0,1)$

Answer 1

The correct option is B $(0,0)$

Let $Q=(a{t}^2,2at)$
Here $a=2$
Let the mid point be $R$
$\therefore (x,y)=(\frac{-2+a{t}^2}{2},\frac{0+2at}{2})$
$\Rightarrow (x,y)=(\frac{-2+2t^2}{2},\frac{0+4t}{2})$
$\Rightarrow (x,y)=(-1+t^2,2t)$
$\Rightarrow x=-1+{\left(\frac{y}{2}\right)}^2$
$\Rightarrow y^2=4(x+1)$
Locus of $R$ is $y^2=4(x+1)$
Here the vertex is $(-1,0)$ and value of $a=1$
Hence focus will be $(-1+1,0)=(0,0)$