If $z_{1}, z_{2}, z_{3}$ be three copmlex numbers whose moduli are $1, 2, 3$ respectively and $| z_{1} + z_{2} + z_{3} | = 1$ . then find the value of $| {9 z}_{1} z_{2} + z_{2} z_{3} + {4 z}_{3} z_{1} |$ .

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Solution

$| z_{1} + z_{2} + z_{3} | = 1$ Squaring
$(z_{1} + z_{2} + z_{3}) (_{1} +_{2} +_{3}) = 1$
${| z_{1} |}_{1}^{2} + {| z_{2} |}_{2}^{2} + {| z_{3} |}_{3}^{2} + z_{1} (_{1} +_{3}) + z_{2} (_{3} +_{1}) + z_{3} (_{1} +_{2}) = 1$
or $\sum z_{1} (_{2} +_{3}) = 1 - (1 + 4 + 9) = - 13$ $(1)$
Now ${| 9 z_{1} z_{2} + z_{2} z_{3} + 4 z_{3} z_{1} |}_{1}^{2} = (9_{1}_{2} +_{2}_{3} + 4_{3}_{1})$
$= 81 (1.4) + 1 (4.9) + 16 (9.1) + 9 {|_{2} |}^{2} . z_{1}_{3} + 36 {|_{1} |}^{2} z_{2} ¯ . z_{3} + 9 {|_{2} |}^{2} z_{3}_{1} + 4 {|_{3} |}^{2} z_{2}_{1} + 36 {|_{1} |}^{2} z_{3}_{2} + 4 {| z_{3} |}_{3}^{2} z_{1}_{2}$
Now put ${|_{1} |}^{2} = 1, {| z_{2} |}_{2}^{2} = 4$ and ${| z_{2} |}_{2}^{2} = 9$
$= 324 + 36 + 144 + 36 \sum z_{1} ({¯ z}_{2} + {¯ z}_{3})$
$= 36 + (9 + 1 + 4) + 36 (- 13) = 36$ by $(1)$
$∴ | 9 z_{1} z_{2} + z_{2} z_{3} + 4 z_{3} z_{1} | = 6$

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Similar questions

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

are three complex numbers whose moduli are

a, b, c

respectively and are such that

∣ ∣ ∣ ∣ \begin{matrix} a & b & c b & c & a c & a & b \end{matrix} ∣ ∣ ∣ ∣ = 0

.
If

z_{1} \neq z_{2}

, then prove that

arg {(\frac{z_{3} - z_{1}}{z_{2} - z_{1}})}^{2} = arg (\frac{z_{3}}{z_{2}})

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

| 9 z_{1} z_{2} + 4 z_{1} z_{3} + z_{2} z_{3} | = 12,

then the value of

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

Q. If

z_{1}

z_{2}

and

z_{3}

are three distinct complex numbers such that

\frac{1}{∣ z_{1} - z_{2} ∣} = \frac{3}{∣ z_{2} - z_{3} ∣} = \frac{5}{∣ z_{1} - z_{3} ∣}

, then the value of

\frac{1}{z_{1} - z_{2}} + \frac{9}{z_{2} - z_{3}} + \frac{25}{z_{3} - z_{1}}

If $z_{1}, z_{2}, z_{3}$ are complex numbers such that $| z_{1} | = | z_{2} | = | z_{3} | = ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ = 1, then find the value of | z_{1} + z_{2} + z_{3} | .$

Q. If

z_{1}, z_{2}

and

z_{3}

are complex numbers such that

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

then

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

is :

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If z1,z2,z3 be three copmlex numbers whose moduli are 1,2,3 respectively and |z1+z2+z3|=1. then find the value of |9z1z2+z2z3+4z3z1|.

If $z_{1}, z_{2}, z_{3}$ be three copmlex numbers whose moduli are $1, 2, 3$ respectively and $| z_{1} + z_{2} + z_{3} | = 1$ . then find the value of $| {9 z}_{1} z_{2} + z_{2} z_{3} + {4 z}_{3} z_{1} |$ .