Let a, b, x and y be real numbers such that a – b = 1 and y \neq 0. If the complex number z = x + iy satisfies Im \textrm{ } \left(\right. \frac{az + b}{z + 1} \left.\right) = y, then which of the following is(are) possible value(s) of x?

Engineering

Mathematics

Conjugate Modulus and Argument of a Complex Number

Question

Let a, b, x and y be real numbers such that a – b = 1 and y $\neq$ 0. If the complex number z = x + iy satisfies $Im (\frac{az + b}{z + 1})$ = y, then which of the following is(are) possible value(s) of x?

$1 - \sqrt{1 + y^{2}}$

$1 + \sqrt{1 + y^{2}}$

$- 1− \sqrt{1 - y^{2}}$

$- 1 + \sqrt{1 - y^{2}}$

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a – b = 1; y $\neq$ 0, a, b, c, y $\in$ R

z = x + iy

$\frac{az + b}{z + 1} = \frac{ax + aiy + b}{x + iy + 1} = \frac{((ax + b) + iay) ((x + 1) - iy)}{{(x + 1)}^{2} + y^{2}}$

$Im (\frac{az + b}{z + 1}) = \frac{ay (x + 1) - (ax + b) y}{{(x + 1)}^{2} + y^{2}} = \frac{ay - by}{{(x + 1)}^{2} + y^{2}} = y$

$∴$ (x + 1)² + y² = 1 ( $∵$ a – b = 1)

⇒ x + 1 = ± $\sqrt{1 - y^{2}} \Rightarrow x = - 1 + \sqrt{1 - y^{2}} and - 1 - \sqrt{1 - y^{2}} .$