Question

if x+iy=√{(a+ib)/(c+id)} , then prove that (x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)

Answer 1

It is given that $x+iy = \sqrt{\frac{a+ib}{c+id}}$ (Equation 1)

We have to prove that $(x^2+y^2)^2 = \frac{a^2+b^2}{c^2+d^2}$

Consider the conjugate of x+iy,

So, $x-iy = \sqrt{\frac{a-ib}{c-id}}$  (Equation 2)

Multiplying equation 1 by 2, we get

$(x+iy)(x-iy) =\sqrt{\frac{a+ib}{c+id}} \sqrt{\frac{a-ib}{c-id}}$

$x^2+y^2 =\sqrt{\frac{(a+ib)(a-ib)}{(c+id)(c-id)}$

$x^2+y^2 =\sqrt{\frac{a^2-(ib)^2}{c^2-(id)^2}}$

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

Squaring on both the sides, we get

$(x^2+y^2)^2 =\sqrt{(\frac{a^2+b^2}{c^2+d^2})}^2$

Therefore, $(x^2+y^2)^2 =(\frac{a^2+b^2}{c^2+d^2})$

Hence, proved.

Answer 2

Step-by-step explanation:

Ans in attachment.

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