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Properties of nth Root of a Complex Number

Points z1 ...

Question

Points $z_{1}$ & $z_{2}$ are adjacent vertices of a regular octagon. The vertex $z_{3}$ adjacent to $z_{2} (z_{3} \neq z_{1})$ can be represented by -

A

z_{2} + \frac{1}{\sqrt{2}} (1 \pm i) (z_{1} + z_{2})

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B

z_{2} + \frac{1}{\sqrt{2}} (- 1 \pm i) (z_{1} - z_{2})

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C

z_{2} + \frac{1}{\sqrt{2}} (- 1 \pm i) (z_{2} - z_{1})

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D

none of these

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Solution

The correct option is D $z_{2} + \frac{1}{\sqrt{2}} (- 1 \pm i) (z_{1} - z_{2})$
Since

$z_{2} = z_{1} e^{\pm \frac{i π}{4}}$

$z_{3} = z_{2} e^{\pm \frac{i π}{4}}$

Now,

$z_{3} - z_{2} = (z_{2} - z_{1}) \frac{(1 \pm i)}{\sqrt{2}}$

$z_{3} = z_{2} + (z_{2} - z_{1}) \frac{(1 \pm i)}{\sqrt{2}}$

$z_{3} = z_{2} + \frac{1}{\sqrt{2}} (- 1 \pm i) (z_{1} - z_{2})$

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Similar questions

z_{1}

z_{2}

are adjacent vertices of a regular octagon. The vertex

z_{3}

z_{2} (z_{3} \neq z_{1})

can be represented by

| z_{1} | = | z_{2} | = | z_{3} | = 1

z_{1}, z_{2}, z_{3}

are represented by the vertices of an equilateral triangle. Then

z_{1}

z_{2}

are adjacent vertices of a regular polygon of n sides. Find the vertex

z_{3}

z_{2} (z_{3} \neq z_{1}) .

z_{1}, z_{2}

z_{3}

are the affixes of three points

A, B

C

respectively and satisfy the condition

| z_{1} - z_{2} | = | z_{1} | + | z_{2} |

| (2 - i) z_{1} + i z_{3} | = | z_{1} | + | (1 - i) z_{1} + i z_{3} |

then prove that

△ A B C

in a right angled.

| z_{2} + i z_{1} | = | z_{1} | + | z_{2} |

| z_{1} | = 3

| z_{2} | = 4

△ A B C

, if affixes of

A, B

C

z_{1}, z_{2}

[\frac{z_{2} - i z_{1}}{1 - i}]

respectively, is,

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Properties of nth Root of a Complex Number

Standard XII Mathematics

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