Question

The line segment joining $A\left(5,0\right)$ and $B\left(10cos\theta ,10sin\theta \right)$ is divided internally in the ratio $2:3$ at $P$ . If $\theta$ varies, then the perimeter of locus of $P$ is

• A. $4\pi$ units
• B. $16\pi$ units
• C. $8\pi$ units
• D. $6\pi$ units

Answer 1

Answer: (C)

Let $P\left(h,k\right)$ be the point dividing te line segment in the ratio $2:3$ , then

$h=\frac{2\left(10cos\theta \right)+3\left(5\right)}{2+3}=4cos\theta +3$ and $k=\frac{2\left(10sin\theta \right)+3\left(0\right)}{2+3}=4sin\theta$
$\therefore {\left(h-3\right)}^2+k^2=16$
$\Rightarrow$ Locus of $P\left(h,k\right)$ is ${\left(x-3\right)}^2+y^2=16$ , which is a circle.
Hence, the perimeter of the circle is $2\pi \left(4\right)=8\pi$ units