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Q.
The line joining $(5,0)$ to $((10 \cos \,\theta, 10 \sin \,\theta)$ is divided internally in the ratio $2: 3$ at $P$. If $\theta$ varies, then the locus of $P$ is
Conic Sections
Solution:
Let $P ( x , y )$ be the point dividing the join of $A$ and $B$ in the ratio $2: 3$ internally, then
$x =\frac{20 \cos \theta+15}{5}=4 \cos \theta+3$
$ \Rightarrow \cos \theta=\frac{ x -3}{4} \,\,\,\,\,\,\,\,...(i)$
$y =\frac{20 \sin \theta+0}{5}=4 \sin \theta$
$ \Rightarrow \sin \theta=\frac{ y }{4} \,\,\,\,\,\,\,\, ....(ii)$
Squaring and adding (i) and (ii), we get the required
locus $(x-3)^{2}+y^{2}=16$, which is a circle.