Let S = z C : z z - (a-2i)z - (a+2i) z + 4 = 0 where a > 0 , and T = z C : z z - (5-6i)z - (5+6i) z + 52 = 0 . If the sets S and T intersect at exactly one point, then the sum of all possible values of a is

Step-by-step solution

The general equation of a circle in the complex plane is z z + z + z + k = 0 , which has center - and radius | |^2 - k . For set S : = -(a-2i) = -(a+2i) Center C₁ = - = a+2i Radius R₁ = |a+2i|^2 - 4 = a^2 + 4 - 4 = a (since a > 0 ). For set T : = -(5-6i) = -(5+6i) Center C₂ = - = 5+6i Radius R₂ = |5+6i|^2 - 52 = 25 + 36 - 52 = 9 = 3 . For the two circles to intersect at exactly one point, they must touch each other either externally or internally. The distance d between their centers must satisfy d = R₁ + R₂ or d =