Let z be a complex number satisfying |z-2|=|z-2i| and Im ( z+2 z-2 )=- 4 5 . If z₁ and z₂ are the possible values of z , then the value of 4(|z₁|^2+|z₂|^2) is equal to

Step-by-step solution

The condition |z-2|=|z-2i| implies that z lies on the perpendicular bisector of the line segment joining (2,0) and (0,2) . This perpendicular bisector is the line y=x . Let z = x + ix = x(1+i) . Substitute z into the given rational expression: z+2 z-2 = x+2+ix x-2+ix To find the imaginary part, multiply the numerator and the denominator by the conjugate of the denominator, which is (x-2)-ix : z+2 z-2 = ((x+2)+ix)((x-2)-ix) (x-2)^2+x^2 The denominator simplifies to x^2 - 4x + 4 + x^2 = 2x^2 - 4x + 4 . The imaginary