The locus of the midpoint of the chords of the circle $x^2+y^2=4$ which subtend an angle of $120^{\circ }$ at the center is a circle. Find the radius of this circle.

Solution

1. Let the midpoint be $(h,k)$.
2. The distance from the origin to the chord is $d=\sqrt{h^2+k^2}$.
3. In the triangle formed by the center and the chord endpoints, $d=R\cos (60^{\circ })$.
4. Given $R=2$, we have $$\begin{array}{rl}d& =2\times (1/2)\\ & =1\end{array}$$.
5. Thus $h^2+k^2=1$.
6. Hence the answer is (A).