Question

if x+iy=u+iv/u-iv,then prove x^2+y^2=1??anyone??

Answer 1

Given:

The given relation is

$x + iy = \frac{u + iv}{u - iv}$

To Prove:

$x^2+y^2 = 1$

Proof:

Using the given relation, we have

$x + iy = \frac{u + iv}{u - iv}$

Taking modulus on both sides, we get

$|x+iy| = |\frac{u+iv}{u-iv}|$

$\sqrt{x^2 + y^2} = \frac{|u+iv|}{|u-iv|}$                         [Using |x+iy| = $\sqrt{x^2+y^2}$]

$\sqrt{x^2 + y^2} = \frac{\sqrt{u^2+v^2} }{\sqrt{u^2+(-v)^2} }$

$\sqrt{x^2+y^2} = \frac{\sqrt{u^2+v^2}} {\sqrt{u^2+v^2}}$

$\sqrt{x^2+y^2} = 1$

Squaring both sides, we get

$x^2+y^2 = 1$

L.H.S. = R.H.S.

Hence proved.

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