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Modulus of a Complex Number

State whether...

Question

State whether the following statement is true or false
The triangle whose varticles are the point represented by the complex number $z_{1}, z_{2}, z_{3}$ on the argand diagram is equilateral if and only if $\frac{1}{z_{2} - z_{3}} + \frac{1}{z_{3} - z_{1}} + \frac{1}{z_{1} - z_{2}} = 0$ that is if ${z_{1}}^{2} + {z_{2}}^{2} + {z_{3}}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$

A

True

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B

False

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Solution

The correct option is A True
$\frac{z_{1} - z_{2}}{l} = \frac{z_{3} - z_{2}}{l} e^{i π / 3}$ ...........[1]

$\frac{z_{2} - z_{3}}{l} = \frac{z_{1} - z_{3}}{l} e^{i π / 3}$ ...........[2]

from [1] and [2]:

$\frac{1}{z_{2} - z_{3}} + \frac{1}{z_{3} - z_{1}} + \frac{1}{z_{1} - z_{2}} = 0$ that is

${z_{1}}^{2} + {z_{2}}^{2} + {z_{3}}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$

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Similar questions

Q. Assertion :Let

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

be three complex numbers such that

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

1 + z_{1} + z_{2} + z_{3} = 0

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

will represent vertices of an equilateral triangle on the complex plane. Reason:

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

represent vertices of an equilateral triangle if

z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}

.

z_{1}, z_{2}

z_{3}

be three complex numbers such that

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

\frac{z_{1}^{2}}{z_{2} z_{3}} + \frac{z_{2}^{2}}{z_{3} z_{1}} + \frac{z_{3}^{2}}{z_{1} z_{2}} + 1 = 0.

Then the sum of all possible values of

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

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

be complex numbers such that

z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}

| z_{1} + z_{2} + z_{3} | = 21.

| z_{1} - z_{2} | = 2 \sqrt{3}, | z_{1} | = 3 \sqrt{3},

then the value of

| z_{2} |^{2} + | z_{3} |^{2}

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

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

are represented by the vertices of an equilateral triangle. Then

Q. The triangle whose vertices are the points represented by the complex numbers

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

on the Argand diagram is equilateral. Then show that

\frac{1}{z_{2} - z_{3}} + \frac{1}{z_{3} - z_{1}} + \frac{1}{z_{1} - z_{2}} = 0

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