Question

$\text{If}x+iy=\sqrt{\frac{a+ib}{c+id}},\text{then write the value of}(x^2+y^2{)}^2.$

Answer 1

$x+iy=\sqrt{\frac{a+ib}{c+id}}...\left(1\right)$

Taking conjugates on both sides we get

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

Multiplying (1) and (2) we get

$(x+iy)(x-iy)=\sqrt{\frac{a+ib}{c+id}}\times \sqrt{\frac{a-ib}{c-id}}\Rightarrow x^2+y^2=\sqrt{\frac{(a+ib)(a-ib)}{(c+id)(c-id)}}\Rightarrow x^2+y^2=\sqrt{\frac{a^2+b^2}{c^2+d^2}}(x^2+y^2{)}^2=\frac{a^2+b^2}{c^2+d^2}$