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 $\theta ϵR$ is

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

Answer 1

Answer: (B)

a circle

Given points $A(1,0)$ and $B(2\cos \theta ,2\sin \theta )$

Equation of line $AB$

$AB:y=\frac{2\sin \theta }{2\cos \theta -1}(x-1)-----\left(1\right)$

Point P divides line $AB$ in ratio $2:3$ internally

By section formula

$P(\frac{4\cos \theta +3}{5},\frac{4\sin \theta }{5})$

Let $h=\frac{4\cos \theta +3}{5}$ and $k=\frac{4\sin \theta }{5}$

Distance of point P from $A(1,0)$ is 2

By distance formula

$(h-1{)}^2+(k{)}^2=4$

Here above equation represents circle