Let a and b be distinct real numbers. Consider the set S = z C : Re (z^2 + iaz) = -b and Re (z^2 + ibz) = -a . If the set S contains exactly one element, then the value of a + b is equal to

Step-by-step solution

Let z = x + iy , where x, y R . We have z^2 + iaz = (x + iy)^2 + ia(x + iy) = x^2 - y^2 + 2ixy + iax - ay . Taking the real part, we get: Re (z^2 + iaz) = x^2 - y^2 - ay The given equations become: x^2 - y^2 - ay = -b (1) x^2 - y^2 - by = -a (2) Subtracting equation (2) from equation (1) gives: (b - a)y = a - b Since a and b are distinct real numbers, a - b 0 . Dividing by b - a yields y = -1 . Substituting y = -1 into equation (1): x^2 - (-1)^2 - a(-1) = -b x^2 - 1 + a = -b x^2 = 1 - a - b For the set S to contain