Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the midpoint of the line segment $P Q$, then locus of $R$ is

• Correct answer
  A

  a circle
• B

  a circle and a pair of straight lines
• C

  a rectangular hyperbola
• D

  a pair of sraight lines

Solution

Let circle be $\left(x-x_{1}\right)=r \cos \theta$ $y-y_{1}=r \sin \theta$ $h=\frac{a+x_{1}+r \cos \theta}{2}$ $k=\frac{b+y_{1}+r \sin \theta}{2}$ $\Rightarrow\left(h-\frac{\left(a+x_{1}\right)}{2}\right)^{2}+\left(k-\frac{\left(b+y_{1}\right)}{2}\right)^{2}=\frac{r^{2}}{4}$

Hence, the correct answer is option: 'A'.