Z1 = x + y + (x ^ 2-xy-2y) I, Z2 = (2x-y) - (y-xy) I, ask when x and y are real numbers (1) Are Z1 and Z2 real numbers? (2) Z1 and Z2 are conjugate complex numbers?

Z1 = x + y + (x ^ 2-xy-2y) I, Z2 = (2x-y) - (y-xy) I, ask when x and y are real numbers (1) Are Z1 and Z2 real numbers? (2) Z1 and Z2 are conjugate complex numbers?

The solution (1) ∵ Z1, Z2 are real numbers, ∵ (x ^ 2-xy-2y) = 0, (y-xy) = 0. The solution is x = 1, y = 1 / 3 or x = y = 0
(2) ∵ Z1, Z2 are conjugate complex numbers ∵ x ^ 2-xy-2y = - [- (y-xy)], that is, x ^ 2-xy-2y = y-xy, and the solution is x = 3 / 2, y = 3 / 4
That's the idea. As a result, you can check it,

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