Question

The locus of the midpoint of a line segment that is drawn from a given external point $P$ to a given circle with center $O$ (where $O$ is origin) and radius $r$, is

• A. A straight line perpendicular to $PO$
• B. A circle with center $P$ and radius $r$
• C. A circle with center $P$ and radius $2r$
• D. A circle with center at the midpoint $PO$ and radius $\frac{r}{2}$

Answer 1

The correct option is D A circle with center at the midpoint $PO$ and radius $\frac{r}{2}$

Let the locus be $(h,k)$ and point $p$ be $(\alpha ,\beta )$
$2h=\alpha +r\cos \theta$
$2k=\beta +r\sin \theta$
$\Rightarrow (2h-\alpha {)}^2+(2k-\beta {)}^2=r^2$
${(h-\frac{\alpha }{2})}^2+{(k-\frac{\beta }{2})}^2={\left(\frac{r}{2}\right)}^2$
Locus is ${(x-\frac{\alpha }{2})}^2+{(y-\frac{\beta }{2})}^2={\left(\frac{r}{2}\right)}^2$
which is a circle with centre as midpoint of OP and radius $\frac{r}{2}$