Question

If |z₁| = 1(z₁ ≠ –1) and z₂ = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z₂ is zero.

Answer 1

Let z₁ = x + yi

|z₁| = `sqrt(x^2 + y^2)` = 1   ......[Given that |z₁| = 1]

⇒ x² + y² = 1  ......(i)

Now z₂ = `(z_1 - 1)/(z_1 + 1)`

= `(x + yi - 1)/(x + yi + 1)`

= `((x + 1) + y"i")/((x + 1) + y"i")`

= `((x - 1) + yi)/((x + 1) + yi) xx (x + 1 - yi)/(x + 1 - yi)`

= `((x - 1)(x + 1) - y(x - 1)i + y(x + 1)i - y^2i^2)/((x + 1)^2 - y^2i^2)`

= `(x^2 - 1 + yi(x + 1 - x + 1) + y^2)/(x^2 + 1 + 2x + y^2)`

= `((x^2 + y^2 - 1) + 2yi)/(x^2 + y^2 + 2x + 1)`

= `((1 - 1))/(x^2 + y^2 + 2x + 1) + (2y)/(x^2 + y^2 + 2x + 1) "i"`

= `0 + (2y)/(x^2 + y^2 + 2x + 1) "i"`

Hence, the real part of z₂ is 0.