Question

Example 2. If z1 and z2 (≠0) are two complex numbers such that |z1 - z2/ z1 + z2| = 1, then
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Answer 1

$\bf \large\underbrace\orange{Question:}$

$\bf \:If \: z_1 \: and \: z_2 \: are \: two \: complex \: numbers \: such \: that$

$\bf \: |\dfrac{z_1 - z_2 }{z_1 + z_2 } | = 1 \: then$

$\bf \large\underbrace\orange{Answer:}$

$\bf\underbrace\orange{Identity \: Used :}$

$\bf \:If |z - z_1| = |z - z_2 |$

$\bf\implies \:z \: lies \: on \: perpendicular \: bisector \:$

$\bf \:of \: A (z_1) \: and \: B (z_2 )$

$\bf\underbrace\orange{Solution:}$

$\bf \: |\dfrac{z_1 - z_2 }{z_1 + z_2 } | = 1 \:$

$\bf\implies \:\dfrac{ |z_1 - z_2 | }{ |z_1 + z_2 | } = 1$

$\bf\implies \: |z_1 - z_2 | = |z_1 + z_2 |$

$\bf\implies \: |\dfrac{z_1}{z_2 } - 1| = |\dfrac{z_1}{z_2 } + 1|$

$\bf\implies \: |\dfrac{z_1}{z_2 } - (1 + 0i)| = |\dfrac{z_1}{z_2 } - ( - 1 + 0i)|$

$\bf\implies \:\dfrac{z_1}{z_2 } \: lies \: on \: perpendicular \: bisector$

$\bf \:of \: line \: segment \: A (-1 + 0i) \: and \: B (1 + 0i)$

$\bf\implies \:\dfrac{z_1}{z_2 } = ki \: where \: k \: ∈ R$

$\bf\implies \:z_1 = ikz_2 \: where \: k∈ R$

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