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

The correct option is B A circle

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 $\frac{(x-3{)}^2}{4^2}+\frac{y^2}{4^2}=1\Rightarrow (x-3{)}^2+y^2=16$, which is a circle.