If the number z+1z−1 is a pure imaginary, then prove that ∣z∣=1.

Let z+1z−1 be purely imaginary. Then (x+1) 2+y2x2+y2−1=0,Whence x2+y2−1=0, that is ∣z∣=x2+y2=1. This completes the proof.

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If the number { z-1 }{ z+1 } is a pure imaginary, then prove

Question

If the number $\frac{z - 1}{z + 1}$ is a pure imaginary, then prove that $∣ z ∣ = 1$ .

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William Discussed

If the number

\frac{z - 1}{z + 1}

is a pure imaginary, then prove that

∣ z ∣ = 1

5 mins ago

Discuss this question LIVE

5 mins ago

Let

\frac{z - 1}{z + 1}

be purely imaginary. Then

\frac{x ^{2} + y ^{2} - 1}{( x + 1 ) ^{2} + y ^{2}} = 0

Whence

x^{2} + y^{2} - 1 = 0

, that is

∣ z ∣ = x^{2} + y^{2} = 1

. This completes the proof.

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Question Text	If the number $\frac{z - 1}{z + 1}$ is a pure imaginary, then prove that $∣ z ∣ = 1$ .
Answer Type	Text solution:1
Upvotes	150