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)

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