- Home
- Standard 11
- Mathematics
9.Straight Line
medium
The locus of a point $P$ which divides the line joining $(1, 0)$ and $(2\cos \theta ,2\sin \theta )$ internally in the ratio $2 : 3$ for all $\theta $, is a
A
Straight line
B
Circle
C
Pair of straight lines
D
Parabola
(IIT-1986)
Solution
(b) Let the coordinates of the point $P$ which divides the line joining $(1, 0)$ and $(2\cos \theta ,\,2\sin \theta )$in the ratio $2:3$ be $(h,k)$. Then, $h = \frac{{4\cos \theta + 3}}{5}$and $k = \frac{{4\sin \theta }}{5}$
==> $\cos \theta = \frac{{5h – 3}}{4}$and $\sin \theta = \frac{{5k}}{4}$
==>${\left( {\frac{{5h – 3}}{4}} \right)^2} + {\left( {\frac{{5k}}{4}} \right)^2} = 1$==>${(5h – 3)^2} + (5{k^2}) = 16$
Therefore locus of $(h,k)$ is ${(5x – 3)^2} + {(5y)^2} = 16$,which is $a$ circle.
Standard 11
Mathematics
