Let z be a complex number. If the system of linear equations (z + i)x + (1 + i)y = 0 (2 - 2i)x + ( z - i)y = 0 has more than one solution, then the maximum value of |z - 3 - 3i| is equal to

Step-by-step solution

For a homogeneous system of linear equations to have more than one solution, the determinant of its coefficient matrix must be zero. vmatrix z + i & 1 + i 2 - 2i & z - i vmatrix = 0 (z + i)( z - i) - (1 + i)(2 - 2i) = 0 We know that (z + i)( z - i) = (z + i)( z + i ) = |z + i|^2 . Also, (1 + i)(2 - 2i) = 2(1 + i)(1 - i) = 2(1^2 - i^2) = 2(2) = 4 . Substituting these into the determinant equation gives: |z + i|^2 - 4 = 0 |z + i| = 2 This represents a circle in the complex plane with center C(0, -1) (or -i ) and radi