Question

The line segment joining $(5,0)$ and $(10\cos \theta ,10\sin \theta )$ 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. a parabola
• E. an ellipse

Answer 1

The correct option is B a circle

Let the coordinates of point $P$ be $(x,y)$. Then,
$\Rightarrow x=\frac{20\cos \theta +15}{5}$
and $y=\frac{20\sin \theta +0}{5}$
$\Rightarrow x=4\cos \theta +3$ and $y=4\sin \theta$
$\Rightarrow x-3=4\cos \theta$ and $y=4\sin \theta$
$\Rightarrow {(x-3)}^2+y^2=16$
which is an equation of a circle.