• # question_answer The locus of a point P which divides the line joining (1, 0) and $(2\cos \theta ,2\sin \theta )$internally in the ratio 2 : 3 for all $\theta$, is a  [IIT 1986] A)            Straight line                               B)            Circle C)            Pair of straight lines    D)            Parabola

Let the coordinates of the point P which divides the line joining (1, 0) and $(2\cos \theta ,\,2\sin \theta )$in the ratio $2:3$ be$(h,k)$. Then,  $h=\frac{4\cos \theta +3}{5}$and $k=\frac{4\sin \theta }{5}$                    Þ $\cos \theta =\frac{5h-3}{4}$and $\sin \theta =\frac{5k}{4}$                    Þ${{\left( \frac{5h-3}{4} \right)}^{2}}+{{\left( \frac{5k}{4} \right)}^{2}}=1$Þ${{(5h-3)}^{2}}+(5{{k}^{2}})=16$ Therefore locus of $(h,k)$is ${{(5x-3)}^{2}}+{{(5y)}^{2}}=16$,which is a circle.