2017Complex NumbersHard

Question

Let a, b, x and y be real numbers such that a - b = 1 and y #0. If the complex number z = x + iy satisfies
Im(az+bz+1) = y, then which of the following is(are) possible value(s) of x ?

Options

A.

-1-1-y2

B.

1+1+y2

C.

1-1+y2

D.

-1+1-y2

Solution

Im(a(x+iy)+bx+iy+1) = y

⇒(x+1)2+ y2 = y⇒x=-1±1-y2

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