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 centre, is       AIEEE  Solved  Paper-2006

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

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

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

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

Correct Answer: C

Solution :

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\[\Delta AOP,\]    \[\cos \frac{\pi }{3}=\frac{OP}{OA}\] \[\Rightarrow \]\[\frac{1}{2}=\frac{\sqrt{{{h}^{2}}+{{k}^{2}}}}{3}\] \[\Rightarrow \]\[{{h}^{2}}+{{k}^{2}}={{\left( \frac{3}{2} \right)}^{2}}\] \[\Rightarrow \]\[{{h}^{2}}+{{k}^{2}}=\frac{9}{4}\] Hence, the required locus is \[{{x}^{2}}+{{y}^{2}}=\frac{9}{4}\]