The perimeter of the locus of the point $P$ which divides the line segment $\overrightarrow{Q A}$ internally in the ratio $1: 2$, where $A=(4,4)$ and $Q$ lies on the circle $x^2+y^2=9$ is

• A

  $8 \pi$
• Correct answer
  B

  $4 \pi$
• C

  $\pi$
• D

  $9 \pi$

Solution

Let $P \equiv(h, k)$
Also $P \equiv\left(\frac{4+6 \cos \theta}{3}, \frac{4+6 \sin \theta}{3}\right) \equiv(h, k)$
$\begin{aligned} & \therefore 3 h-4=6 \cos \theta \text { and } 3 k-4=6 \sin \theta \\ & \Rightarrow(3 h-4)^2+(3 k-4)^2=36 \\ & \Rightarrow\left(h-\frac{4}{3}\right)^2+\left(k-\frac{4}{3}\right)^2=4 \end{aligned}$ $\text { Perimeter }=2 \pi r=2 \pi \times 2=4 \pi \text {. }$

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