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Trigonometric Ratios of Allied Angles

If z1, z2 a...

Question

If $z_{1}, z_{2}$ are two complex numbers such that $∣ ∣ ∣ \frac{z_{1} - z_{2}}{z_{1} + z_{2}} ∣ ∣ ∣ = 1$ and $i z_{1} = k z_{2}$ where $k \in R$ , the the angle between $z_{1} - z_{2}$ and $z_{1} + z_{2}$ is

A

{tan}^{- 1} (\frac{2 k}{k^{2} + 1})

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B

{tan}^{- 1} (\frac{2 k}{1 - k^{2}})

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C

- 2 {tan}^{- 1} k

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D

2 {tan}^{- 1} k

.

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Solution

The correct options are
A ${tan}^{- 1} (\frac{2 k}{k^{2} + 1})$
D $2 {tan}^{- 1} k$ .

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Similar questions

z_{1}, z_{2}

are two complex numbers such that

∣ ∣ ∣ \frac{z_{1} - z_{2}}{z_{1} + z_{2}} ∣ ∣ ∣ = 1

t z_{1} = k z_{2}

k \in R

, the angle between

(z_{1} - z_{2})

(z_{1} + z_{2})

z_{1}

z_{2}

are two complex numbers such that

∣ ∣ ∣ \frac{z_{1} - z_{2}}{z_{1} + z_{2}} ∣ ∣ ∣ = 1

\frac{i z_{1}}{z_{2}} = k

, where k is a real number. Find the angle between the lines from the origin to the points

z_{1} + z_{2}

z_{1} - z_{2}

z_{1}, z_{2}

are two complex numbers and

c > 0

{| z_{1} + z_{2} |}^{2} \leq (1 + c) {| z_{1} |}^{2} + k {| z_{2} |}^{2},

k =

z_{1}

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are two complex numbers such that

| z_{1} | < 1 < | z_{2} |

∣ ∣ ∣ \frac{1 - z_{1}_{2}}{z_{1} - z_{2}} ∣ ∣ ∣ < 1

.

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are complex numbers such that

| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ ∣ = 1

| z_{1} + z_{2} + z_{3} |

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