Question

if z₁,z₂,z₃ are distinct complex no. such that 3/|z₁|=4/|z₂|=5/|z₃| such that |9/z₁+16/z₂+25/z₃|=K|z₁+z₂+z₃|/|z₁|² Find K

Answer 1

$\huge{\text{\underline{Solution:-}}}$

★ Given:-

• | z₁ | = | z₂ | = | z₃ | = 1 ............... (1)
• | 1/z₁ + 1/z₂ + 1/z₃ | = 1 ............ (2)

★ To Find:-

• | z₁ + z₂ + z₃

$\huge{\text{\underline{Explaination:-}}}$

★ We know that:-

• (i) | z | = | z' |, ... z' is conjugate of z
• (ii) z·z' = | z |² = | z' |²
• (iii) Σ(z') = [ Σ(z) ]'

★ Further information:-

If | z | = 1, then : z·z' = 1

i.e., (1/z) = z'.......... (3)

∴ | z₁ + z₂ + z₃ |

= | ( z₁ + z₂ + z₃ )' |

= | (z₁)' + (z₂)' + (z₃)' |

= | 1/z₁ + 1/z₂ + 1/z₃ | ........ from (3)

= 1 ........................ from (2)

∴ | z₁ + z₂ + z₃ | = 1

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