question_answer

Let C be the circle with centre \[(0,0)\] and radius 3 units. 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 center is

A) \[{{x}^{2}}+{{y}^{2}}=\frac{3}{2}\]              

B) \[{{x}^{2}}+{{y}^{2}}=1\]

C) \[{{x}^{2}}+{{y}^{2}}=\frac{27}{4}\]            

D) \[{{x}^{2}}+{{y}^{2}}=\frac{9}{4}\]

Correct Answer: D

Solution :

Let \[M(h,k)\] be the mid-point of chord AB where

\[\angle ACB=\frac{2\pi }{3}\]          \[\therefore \angle ACM=\frac{\pi }{3}\]

Also  \[CM=3\,\cos \frac{\pi }{3}=\frac{3}{2}\]

\[\Rightarrow \,\sqrt{{{h}^{2}}+{{k}^{2}}}=\frac{3}{2}\Rightarrow {{h}^{2}}+{{k}^{2}}=\frac{9}{4}\]

\[\therefore \] Locus of \[(h,k)\] is \[{{x}^{2}}+{{y}^{2}}=\frac{9}{4}\]