Playing with Geometer's Sketchpad shows the locus as a circle with half the linear size of either original circle. The diagram (click here) shows the circle with center A, and the arbitrary point is labeled B, but it the K mentioned in the problem. That point was chosen near the circumference of the circle centered at A so it would be obveious that the circle is not an ellipse with positive eccentricity. When this point was closer to the center, the circle was the same size, but one might have thought it was an ellipse of some small positive eccentricity.