Q. 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 :
AIEEEAIEEE 2008
Solution:
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 $\Delta$O PA,
$cos \frac{\pi}{3}=\frac{OP}{OA}$
$\Rightarrow \, \frac{1}{2}=\frac{\sqrt{h^{2}+k^{2}}}{3}$
$\Rightarrow \, h^{2}+k^{2}=\left(\frac{3}{2}\right)^{2}$
$\Rightarrow \, h^{2}+k^{2}=\frac{9}{4}$
Thus the required locus is
$x^{2}+y^{2}=\frac{9}{4}$
