If x-iy- a-ib a-ib prove that x 2 - y 2 -1

x+iy=a+iba ib ⇒ (x+iy)(x iy)=(a+ib)(a ib)×(a ib)(a+ib) ⇒ x^2 (iy)^2=a^2 (ib)^2a^2 (ib)^2 ⇒ x^2+y^2=a^2+b^2a^2+b^2=1 . ...

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Algebra of Complex Numbers

If x+iy= a+...

Question

If $x + i y = \frac{a + i b}{a - i b}$ prove that $x^{2} + y^{2} = 1$

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Solution

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x + i y = \frac{a + i b}{a - i b}

x + i y = \frac{a + i b}{a - i b}

x^{2} + y^{2} = 1

a + i b = \frac{x + i y}{x - i y}

a^{2} + b^{2} = 1

\frac{b}{a} = \frac{2 x y}{x^{2} - y^{2}}

x - i y = \frac{a - i b}{c - i d}

x^{2} + y^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}

x - i y = \sqrt{\frac{a - i b}{c - i d}}

x^{2} - y^{2} = \frac{a c + b d}{c^{2} + d^{2}}

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