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Complex Numbers — Maths STD 11 Science — Question

CBSE BoardEnglish MediumSTD 11 ScienceMathsComplex Numbers5 Marks
Question
If $z_1, z_2$​​​​​​​ are two complex numbers such that $|\text{z}_1|=|\text{z}_2|$ and $\text{arg(z}_1)+\text{arg(z}_2)=\pi,$ then show that $\text{z}_1=-\bar{\text{z}}_2.$
✓

Answer

$|\text{z}_1|=|\text{z}_2|$
Let $\text{arg(z}_1)=\theta$
$\therefore\text{arg(z}_2)=\pi-\theta$
In polar form, $\text{z}_1=|\text{z}_1|(\cos\theta+\text{i}\sin\theta) \ ...(\text{i})$
$\text{z}_2=|\text{z}_2|\big(\cos(\pi-\theta)+\text{i}\sin(\pi-\theta\big)$
$=|\text{z}_2|\big(-\cos\theta+\text{i}\sin\theta\big)$
$=-|\text{z}_2|\big(\cos\theta-\text{i}\sin\theta\big)$
Finding conjugate of
$\bar{\text{z}}_2=-|\text{z}_2|\big(\cos\theta-\text{i}\sin\theta\big) \ ...(\text{ii})$
(i), (ii) is equal to
$\frac{\text{z}_1}{\text{z}_2}=-\frac{|\text{z}_1|(\cos\theta+\text{i}\sin\theta)}{|\text{z}_2|(\cos\theta+\text{i}\sin\theta)}$
$\frac{\text{z}_1}{\text{z}_2}=-\frac{|\text{z}_1|}{|\text{z}_2|} \ [\because|\text{z}_1|=|\text{z}_2|]$
$\frac{\text{z}_1}{\text{z}_2}=-1$
$\text{z}_1=\bar{-\text{z}}_2$
Hence proved.

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