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Purely Imaginary

If |Z1|+|Z2|+...

Question

If $| Z_{1} | + | Z_{2} | + | Z_{3} | = | Z_{1} + Z_{2} + Z_{3} |$ , then $\frac{Z_{1} Z_{2}}{Z_{3}^{2}} + \frac{Z_{2} Z_{3}}{Z_{2}^{2}} + \frac{Z_{3} Z_{1}}{Z_{2}^{2}}$ is

A

real number

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B

Purely imaginary

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C

of modulus 1

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D

None of these

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Solution

The correct option is A real number
$| z_{1} + z_{2} + z_{3} | = | z_{1} | + | z_{2} | + | z_{3} |$
$⟹$ $z_{1} + z_{2} + z_{3}$ are collinear
And $z_{1} + z_{2} + z_{3}$ lie on same side of origin.
$z_{1} + z_{2} + z_{3}$ have same argument $\frac{z_{1} z_{2}}{z_{3}^{2}} = \frac{r_{1} C i s θ r_{2} C i s θ}{r_{3}^{2} {C i s}^{2} θ} = \frac{r_{1} r_{2}}{r_{3}} e t c$
$∴ \frac{z_{1} z_{2}}{z_{3}^{2}} + \frac{z_{1} z_{3}}{z_{1}^{2}} + \frac{z_{3} z_{1}}{z_{2}^{2}}$ is real no.

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Similar questions

| 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}, 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}

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

Then the possible value(s) of

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

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} |

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}

.

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