Question

The locus of a point P which divides the line joining (1, 0) and $(2cos\theta ,2sin\theta )$ internally in the ratio 2:3 for all $\theta$, is a

• A. Straight line
• B. Circle
• C. Pair of straight lines
• D. Parabola

Answer 1

The correct option is B Circle

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