Let A = z C : |z - 3| + |z + 3| = 10 and B = z C : |z - i| = r , where r > 0 . If the set A B contains exactly 3 elements, then the value of r is

Step-by-step solution

The set A represents an ellipse with foci at ( 3, 0) . The sum of focal distances is 2a = 10 , so the semi-major axis is a = 5 . The distance from the center to a focus is c = 3 . The semi-minor axis b is given by b^2 = a^2 - c^2 = 25 - 9 = 16 , so b = 4 . The Cartesian equation of the ellipse is x^2 25 + y^2 16 = 1 . The set B represents a circle centered at (0, 1) with radius r . Its equation is x^2 + (y - 1)^2 = r^2 . For the circle and ellipse to have exactly 3 points of intersection, the circle must be tangent