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If Z1 and Z2 are two distinct complex numbers such that mod of Z1=mod of Z2. Real part of Z1>0, IM(Z2)

If Z1 and Z2 are two distinct complex numbers such that mod of Z1=mod of Z2. Real part of Z1>0, IM(Z2)<0 then Z1+Z2/Z1-Z2=?

Grade:11

1 Answers

Latika Leekha
askIITians Faculty 165 Points
7 years ago
Given that |z1| = |z2|
Now consider (z1 + z2) / (z1 - z2) . (\overline{z}1 - \overline{z}2) / (\overline{z}1 - \overline{z}2)
= (z1\overline{z}1 – z1\overline{z}2 + z2 \overline{z}1 – z2 \overline{z}2) / |(z1 - z2|2
Simplifying this we get,
(z2 \overline{z}1 – z1\overline{z}2) / |(z1 - z2)|2 (Using the fact that |z1|2 = |z2|2)
We know that z – \overline{z} = x + iy – x + iy = 2iy = 2i Im (z)
Hence, the numerator z2 \overline{z}1 – z1\overline{z}2 becomes = 2i Im (z2 \overline{z}1)
Hence, (z1 + z2) / (z1 - z2) = 2i Im (z2 \overline{z}1) / |(z1 - z2)|2
Which shows that the term is either purely imaginary or zero.
Thanks & Regards
Latika Leekha
askIITians Faculty

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