Q. Let $C$ be the circle with centre (0, 0) and radius 3 unit. The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $\frac{2\pi}{3}$ at its centre, is
A) $x^{2} + y^{2} = 1$
B) $x^{2} + y^{2} = \frac{27}{4}$
C) $x^{2} + y^{2} = \frac{9}{4}$
D) $x^{2} + y^{2} = \frac{3}{2}$
Explanation
(c)
Let the coordinates of a point $P$ be $(h,\ k)$ which is mid point of the chord $AB$
Now, $OP = \sqrt{(h - 0)^{2} + (k - 0)^{2}}$
$= \sqrt{h^{2} + k^{2}}$
In $\mathrm{\Delta}AOP,\ \cos\frac{\pi}{3} = \frac{OP}{OA}$
[image:11_image31.emf]
$\Rightarrow \ \ \frac{1}{2} = \frac{\sqrt{h^{2} + k^{2}}}{3}$
$\Rightarrow \ \ \ h^{2} + k^{2} = \frac{9}{4}$
Hence, the required locus is
$x^{2} + y^{2} = \frac{9}{4}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.