Question

The line joining $(5,0)$ to $(10\cos \theta ,10\sin \theta )$ is divided internally in the ratio $2:3$ at $P$. Then the locus of $P$ is

• A. a pair of straight lines
• B. a straight line
• C. a circle
• D. a parabola

Answer 1

The correct option is C a circle

$P$ is a point which divides line joining points $(5,0)$ and $(10\cos \theta ,10\sin \theta )$ in the ratio $2:3$

Let $P=(h,k)$

$⟹h=\frac{2\times 10\cos \theta +3\times 5}{5}$
$⟹h=4\cos \theta +3$
$⟹\cos \theta =\left(\frac{h-3}{4}\right)$
Also, $k=\frac{3\times 0+2\times 10\sin \theta }{5}$

$⟹k=4\sin \theta$
$⟹\sin \theta =\frac{k}{4}$
$⟹{\left(\frac{h-3}{4}\right)}^2+{\left(\frac{k}{4}\right)}^2=1$
$⟹(h-3{)}^2+k^2=4^2$
$\therefore (x-3{)}^2+y^2=4^2$ is the required locus of $P$ which is the equation of a circle.