Question

From a point Q on the circle $x^2+y^2=a^2$ perpendicular QM are drawn to x-axis find the locus of point 'P' dividing QM in ratio 2 : 1

Answer 1

Let $Q\equiv (a\cos \theta ,a\sin \theta )M\equiv (a\cos \theta ,0)$
Let $P\equiv (h,k)\therefore h=a\cos \theta ,k=\frac{a\sin \theta }{3}$
$\therefore {\left(\frac{3k}{a}\right)}^2+{\left(\frac{h}{a}\right)}^2=1\Rightarrow$ Locus of P is $\frac{x^2}{a^2}+\frac{y^2}{(a/3{)}^2}=1$