If z 1- z 2- z 3 are complex numbers such that - z 1- z 2- z 3-1-z1-1-z2-1- z 3-1- then find the value of - z 1- z 2- z 3-

Given : |z_1|=|z_2|=|z_3|=1 ⇒ |z_1|^2=|z_2|^2=|z_3|^2=1 ⇒ z_1z_1=z_2z_2=z_3z_3=1 ⇒ z_1=1/z_1,z_2=1/z_2,z_3=1/z_3 Given that|1/z_1+1/z_2+1/z_3|=1 ⇒ |z_1+z_2+z_3| ...

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Byju's Answer

Standard XII

Mathematics

Properties of Modulus

If z 1, z 2, ...

Question

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

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Solution

Given :

$| z_{1} | = | z_{2} | = | z_{3} | = 1 \Rightarrow | z_{1} |^{2} = | z_{2} |^{2} = | z_{3} |^{2} = 1 \Rightarrow z_{1}_{1} = z_{2}_{2} = z_{3}_{3} = 1 \Rightarrow_{1} = \frac{1}{z_{1}},_{2} = \frac{1}{z_{2}},_{3} = \frac{1}{z_{3}} Given that ∣ ∣ \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} ∣ ∣ = 1 \Rightarrow |_{1} +_{2} +_{3} | = 1 i . e . | ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ z_{1} + z_{2} + z_{3} | = 1 \Rightarrow | z_{1} + z_{2} + z_{3} | = 1$

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Similar questions

Q. 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

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

is:

Q. If

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

are complex number 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} |

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 :

Q. If

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

be complex number 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} |

If $z_{1}, z_{2} a n d z_{3}$ 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 | $z_{1}$ + $z_{2}$ + $z_{3}$ |

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If z1,z2,z3 are complex numbers such that |z1|=|z2|=|z3|=∣∣1z1+1z2+1z3∣∣=1,then find the value of |z1+z2+z3|.

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