Question

Let $x-iy=\sqrt{\frac{a-ib}{c-id}}$ then prove that $(x^2+y^2{)}^2=\frac{a^2+b^2}{c^2+d^2}$.

Answer 1

Given,

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

Now squaring both sides we get,

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

Now taking modulus both sides we get,

$|x-iy{|}^2=\left|\frac{a-ib}{c-id}\right|$

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

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