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

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

NTA AbhyasNTA Abhyas 2020Straight Lines

Solution:

Let $P\left(h , k\right)$ be the point dividing te line segment in the ratio $2:3$ , then
Solution
$h=\frac{2 \left(10 cos \theta \right) + 3 \left(5\right)}{2 + 3}=4cos ⁡ \theta +3$ and $k=\frac{2 \left(10 sin \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