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Question

Let $$C$$ be the circle with centre $$\left ( 0,0 \right )$$ and radius $$3$$ units. The equation of the locus of the mid points of chord of the circle $$C$$ that subtend an angle of $$\displaystyle \dfrac {2\pi }3$$ at its centre is


A
x2+y2=32
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B
x2+y2=1
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C
x2+y2=274
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D
x2+y2=94
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Solution

The correct option is D $$x^{2}+y^{2}= \displaystyle \frac{9}{4}$$
In $$\triangle CPB,$$

$$\displaystyle \cos \frac{\pi}{3} =\frac{CP}{CB}=\frac{\sqrt{h^{2}+k^{2}}}{3}\Rightarrow h^{2}+k^{2}=\frac{9}{4} $$

$$\displaystyle \therefore $$ Required locus is $$\displaystyle x^{2}+y^{2}=\frac{9}{4}$$

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Mathematics

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