The coordinate of a circle

Question

Suppose two circles intersect and form three regions A, B, and C. The center of circle A is (2,2) and the center of circle B is (x,y). The three regions formed by the two circles are equal in area. What are the coordinates of circle B?

Answer 1

The question doesn't seem to give enough information for a unique solution.

It is possible to show, quite trivially, that the two circles must be of the same radius.

The area of the overlap is twice the area of a segment. Let the segment subtend an angle $\theta$ at the centre of a circle. The area of a segment is $\frac 12r^2(\theta - \sin\theta)$. The overlapping area will be twice this. It will also be equal to $\frac 12\pi r^2$, since the overlapping area plus the non overlapping area of a single circle have to add up to the area of a single circle and they are also equal.

We then have to solve $\theta - \sin\theta = \frac 12 \pi$. This is a transcendental equation for which only an approximate root can be found numerically, and $\theta \approx 2.30988$ in radian measure.

We can then determine the distance between the centres of the circles as $2r\cos\frac 12\theta \approx 0.808r$.

From this, you can work out that the centre of circle $B$ can lie on the locus of a circle of distance $0.808r$ from the centre of circle $A$.

You can write the locus of the centre of $B$ as: $(x-2)^2 + (y-2)^2 = (0.808r)^2$, but without being given more information like the radius of the circles, you can't go further.