Let S be the set of all complex numbers z that satisfy the equation z( z + 3 - i) + k(3 + i) = 0 for some real number k . If z₀ S is the complex number with the maximum modulus, then the value of |z₀ - 2i|^2 is equal to:

Step-by-step solution

Let z = x + iy , then z = x - iy . Substitute z into the given equation: (x + iy)(x - iy + 3 - i) + k(3 + i) = 0 x^2 + y^2 + 3x - ix + 3iy + y + 3k + ik = 0 Separating the real and imaginary parts, we get: Real part: x^2 + y^2 + 3x + y + 3k = 0 Imaginary part: -x + 3y + k = 0 k = x - 3y Substitute k into the real part equation to find the locus of z : x^2 + y^2 + 3x + y + 3(x - 3y) = 0 x^2 + y^2 + 6x - 8y = 0 This represents a circle with center C(-3, 4) and radius r = (-3)^2 + 4^2 - 0 = 5 . The circle passes throu