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Question
If `(z - 1)/(z + 1)` is purely imaginary number (z ≠ – 1), then find the value of |z|.
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Solution
Given that `(z - 1)/(z + 1)` is purely imaginary number.
Let z = x + yi
∴ `(x + yi - 1)/(x + yi + 1) = ((x - 1) + iy)/((x + 1) + iy)`
= `((x - 1) + iy)/((x + 1) + iy) xx ((x + 1) - iy)/((x + 1) - iy)`
⇒ `((x - 1)(x + 1) - iy(x - 1) + (x + 1)iy - i^2y^2)/((x + 1)^2 - i^2y^2)`
⇒ `(x^2 - 1 + iy(x + 1 - x + 1) + y^2)/(x^2 + 1 + 2x + y^2) = (x^2 + y^2 - 1 + 2yi)/(x^2 + y^2 + 2x + 1)`
⇒ `(x^2 + y^2 - 1)/(x^2 + y^2 + 2x + 1) + (2y)/(x^2 + y^2 + 2x + 1)"i"`
Since, the number is purely imaginary, then real part = 0
∴ `(x^2 + y^2 - 1)/(x^2 + y^2 + 2x + 1)` = 0
⇒ x2 + y2 – 1 = 0
⇒ x2 + y2 = 1
⇒ `sqrt(x^2 + y^2)` = 1
∴ |z| = 1
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