If x + iy = a+ib/a-ib prove that x^2 + y^2 = 1

Question

If x + iy = \(\frac{a+ib}{a-ib}\) prove that x² + y² = 1

Answer 1

Given:

⇒ x + iy = \(\frac{a+ib}{a-ib}\)

We know that for a complex number Z=a+ib it’s magnitude is given by |Z| = \(\sqrt{a^2+b^2}\)

We know that \(|\frac{a}{b}|\) is \(|\frac{a}{b}|\) 

Applying Modulus on both sides we get,

Squaring on both sides

⇒ \((\sqrt{x^2+y^2)^2}\) = 1²

⇒ x² + y² = 1 

∴ Thus Proved.