First slide

Equation of line in Straight lines

Question

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

Moderate

A

straight line
B

circle
C

pair of straight lines
D

parabola

Solution

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

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