Question

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

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

Answer 1

The correct option is B

Circle

Step 1: Form the equations by using the section formula.

Assume that, the coordinates of the given point $P$ is $\left(x,y\right)$.

If a point is $(x,y)$ divides a line formed by joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in $m:n$ internally then $x=\frac{m{x}_1+n{x}_2}{m+n}$ and $y=\frac{m{y}_1+n{y}_2}{m+n}$.

According to the section formula.

$x=\frac{4\cos \theta +3}{5}...\left(1\right)$

Also, $y=\frac{4\sin \theta }{5}...\left(2\right)$

Solve the equation $\left(1\right)$ for $\cos \theta$ as follows:

$$\begin{gathered} x=\frac{4\cos \theta +3}{5} \\ \Rightarrow 5x=4\cos \theta +3 \\ \Rightarrow 5x-3=4\cos \theta \\ \Rightarrow \frac{5x-3}{4}=\cos \theta ...\left(3\right) \end{gathered}$$

Step 2: Solve the equations to get the shape of the locus.
Solve the equation $\left(2\right)$ for $\sin \theta$ as follows:

$\frac{5y}{4}=\sin \theta ...\left(4\right)$

Square and add equation $\left(3\right)$ and equation $\left(4\right)$.

$$\begin{gathered} \frac{{(5x-3)}^2}{16}+\frac{25y^2}{16}={\cos }^2\theta +{\sin }^2\theta \\ \Rightarrow {(5x-3)}^2+25y^2=16 \end{gathered}$$

This is the equation of a circle.

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

Hence, the correct answer is option B.