Question

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

• A. a pair of straight lines
• B. a circle
• C. a straight line
• D. None of these

Answer 1

Answer: (B)

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.