Let z be a complex number. The system of equations in variables u and v 2u + (1 + i 3 )v = 2z (1 - i 3 )u + 2v = 2 z is known to have at least one solution. If z also satisfies the condition |z - (2 3 +1) - i(2- 3 )| = 2 , then the value of |z|^2 is equal to

Step-by-step solution

First, evaluate the determinant of the coefficient matrix of the given system: = (2)(2) - (1 + i 3 )(1 - i 3 ) = 4 - (1^2 - (i 3 )^2) = 4 - (1 + 3) = 0 Since = 0 and the system has at least one solution, it must be consistent and have infinitely many solutions. Thus, _u = 0 and _v = 0 . _u = vmatrix 2z & 1 + i 3 2 z & 2 vmatrix = 4z - 2 z (1 + i 3 ) = 0 2z = z (1 + i 3 ) Let z = x + iy . Then: 2(x + iy) = (x - iy)(1 + i 3 ) = x + ix 3 - iy + y 3 2x + i(2y) = (x + y 3 ) + i(x 3 - y) Equating the real and imaginary p