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 + b/z + 1) = y, then which of the following is(are) possible value(s) of x ?

(A)  -1 + √(1 - y²)

(B)  -1 - √(1 - y²)

(C)   1 + √(1 + y²)

(D)   1 - √(1 + y²)

Answer 1

Correct options (A) and (B)

It is given that z = x + iy satisfies Im(az + b/z + 1) = y

Therefore,

Rationalising the above equation: Multiplying and dividing LHS by (x + 1 - iy), we get

Using a² - b² = (a + b)(a - b), we get

Rearranging LHS, we get

ay - by/(x + 1)² + y² (as Im of the value in bracket is coefficient of i)

⇒ -y (a - b) = y(( x + y)²  ⇒(a - b) = (x + 1)² + y²

It is given that a - b = 1 and y ≠ 0. Therefore,