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If |z| = 1, z i, then z can be written in the form

Question

If $∣ z ∣ = 1, z \neq = i$ , then $z$ can be written in the form

\frac{1 + x}{1 - x} (x \in R)

\frac{1 + i x}{1 - i x} (x \in R)

\frac{i + x}{1 - x} (x \in R)

None of these

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Eleanor Discussed

∣ z ∣ = 1, z \neq = i

, then

z

can be written in the form

6 mins ago

Discuss this question LIVE

6 mins ago

Text solutionVerified

∣ z ∣ = 1

Let

z = cos A + i sin A

\Rightarrow \frac{z + 1}{z - 1} = - (\frac{1 + cos A + i sin A}{1 - cos A - i sin A}) = - ⎝ ⎛ \frac{2 cos ^{2} \frac{A}{2} + 2 i sin \frac{A}{2} cos \frac{A}{2}}{2 sin ^{2} \frac{A}{2} - 2 i sin \frac{A}{2} cos \frac{A}{2}} ⎠ ⎞

\Rightarrow \frac{z + 1}{z - 1} = - i cot \frac{A}{2} ⎝ ⎛ \frac{cos \frac{A}{2} + i sin \frac{A}{2}}{i sin \frac{A}{2} - i ^{2} cos \frac{A}{2}} ⎠ ⎞ = - i cot \frac{A}{2} = \frac{1}{i tan \frac{A}{2}}

let

x = tan \frac{A}{2}

\Rightarrow \frac{z + 1}{z - 1} = \frac{1}{i x}

\Rightarrow \frac{2 z}{2} = \frac{1 + i x}{1 - i x}

\Rightarrow z = \frac{1 + i x}{1 - i x}

Ans: B

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Question Text	If $∣ z ∣ = 1, z \neq = i$ , then $z$ can be written in the form
Answer Type	Text solution:1
Upvotes	150