Question

LetC 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 :

• A. $x^{2}+y^{2}=1$
• B. $x^{2}+y^{2}=\frac{27}{4}$
• C. $x^{2}+y^{2}=\frac{9}{4}$
• D. $x^{2}+y^{2}=\frac{3}{2}$

Answer 1

Answer: (C)

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

Answer 2

Let the co-ordinates of a point P be (h, k) which is mid point of the chord AB. $op=\sqrt{\left(h-0\right)^{2}+\left(k-0\right)^{2}}$ $=\sqrt{h^{2}+k^{2}}$ Now in $\Delta OPA$, $cos \frac{\pi}{3}=\frac{OP}{OA}$ $\Rightarrow\, \frac{1}{2}=\frac{\sqrt{h^{2}+k^{2}}}{3}$ $\Rightarrow\, h^{2}+k^{2}=\left(\frac{3}{2}\right)^{2}$ $\Rightarrow\, h^{2}+k^{2}=\frac{9}{4}$ Thus the required locus is $x^{2}+y^{2}=\frac{9}{4}$