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Prove that equation of perpendicular bi...

Prove that equation of perpendicular bisector of line segment joining complex numbers `z_(1)` and `z_(2)` is `z(barz_(2) - barz_(1)) + barz (z_(2) + z_(1)) + |z_(1)|^(2) -|z_(2)|^(2) =0`

Text Solution

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Let z be the complex number of any point on the perpendicular bisector.

Then PA = PB
`rArr = |z- z_(1)| = |z- z_(2)|`
`rArr |z- z_(1)|^(2)=|z-z_(2)|^(2)`
`rArrr |z|^(2) + |z_(1)|^(2) -zbarz_(1) - zbarz_(1) = |z|^(2)+|z_(2)|^(2) - zbarz_(2) - z_(2)barz`
`rArr z (barz_(2)- barz_(1)) + barz (z_(2) - z_(1)) + |z_(1)|^(2) - |z_(2)|^(2) = 0`
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