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 \[{{x}^{2}}+{{y}^{2}}=\frac{9}{4}\]You need to login to perform this action.
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