# Let $\dpi{100} C$ be the circle with centre (0, 0) and radius 3 units. The equation of the locus of the mid points of chord of the circle $\dpi{100} C$        that subtend an angle of  $\dpi{100} 2\pi /3$  at its centre is Option 1) $x^{2}+y^{2}=\frac{3}{2}$ Option 2) $x^{2}+y^{2}=1$ Option 3) $x^{2}+y^{2}=\frac{27}{4}$ Option 4) $x^{2}+y^{2}=\frac{9}{4}$

As we learnt in

Equation of a circle -

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

- wherein

Circle with centre $\left ( O,O \right )$ and radius $r$.

In $\Delta AOB$

$cos\frac{\pi}{3}=\frac{OB}{OA}$

$\sqrt{\frac{h^{2}+k^{2}}{3}}=\frac{1}{2}$

h2 + k2 = 9/4

Option 1)

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

This is incorrect option

Option 2)

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

This is incorrect option

Option 3)

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

This is incorrect option

Option 4)

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

This is correct option

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