The line joining $$\left( {5,\,0} \right)$$  to $$\left( {10\,\cos \,\theta ,\,10\,\sin \,\theta } \right)$$    is divided internally in the ratio $$2 : 3$$  at $$P$$. If $$\theta $$ varies, then the locus of $$P$$ is :

Solution :
Let $$P\left( {x,\,y} \right)$$  be the point dividing the join of $$A$$ and $$B$$ in the ratio $$2 : 3$$  internally, then
$$\eqalign{ & x = \frac{{20\,\cos \,\theta + 15}}{5} = 4\,\cos \,\theta + 3 \cr & \Rightarrow \cos \,\theta = \frac{{x - 3}}{4}......\left( {\text{i}} \right) \cr & y = \frac{{20\,\sin \,\theta + 0}}{5} = 4\,\sin \,\theta \cr & \Rightarrow \sin \,\theta = \frac{y}{4}......\left( {{\text{ii}}} \right) \cr} $$
Squaring and adding $$\left( {{\text{i}}} \right)$$ and $$\left( {{\text{ii}}} \right),$$ we get the required locus $${\left( {x - 3} \right)^2} + {y^2} = 16,$$    which is a circle.