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Q.

$Let C be the circle with centre (0, 0)$ $and radius 3 units. The equation of$ $the locus of the mid points of the$ $chords of the circle C that subtend$ $an angle of \frac{2 π}{3} at its center is$

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a

$x^{2} + y^{2} = \frac{3}{2}$

b

$x^{2} + y^{2} = \frac{9}{4}$

c

$x^{2} + y^{2} = 1$

d

$x^{2} + y^{2} = \frac{27}{4}$

answer is D.

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Detailed Solution

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$\begin{array}{l} Let M (h, k) be the mid-point of chord AB where \\ ∠ A C B = \frac{2 π}{3} \\ ∴ ∠ A C M = \frac{π}{3} \\ Also C M = 3 \cos \frac{π}{3} = \frac{3}{2} \\ \Rightarrow \sqrt{h^{2} + k^{2}} = \frac{3}{2} \Rightarrow h^{2} + k^{2} = \frac{9}{4} \\ ∴ Locus of (h, k) is x^{2} + y^{2} = \frac{9}{4} \end{array}$

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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 2π3 at its center is