Question

If $O$ is the origin and $Q$ is a variable point on the locus $x^2=4y$. Find the equation of the locus of the midpoint of seg $OQ$.

Answer 1

Since $M$ is the midpoint

$\frac{a+0}{2}=h$ $\frac{b+0}{2}=k$

$a=2h$ $b=2k$

Since $(a,b)$ lie on $x^2=4y$

$(2h{)}^2=4(2k)$

$4h^2=8k$

$h^2=2k\Rightarrow x^2=2y$.