Question

If x + iy =\[\sqrt{\frac{a + ib}{c + id}}\] then write the value of (x² + y²)².

Answer 1

\[x + iy = \sqrt{\frac{a + ib}{c + id}}\]

\[\text { Taking modulus on both the sides }, \]

\[\left| x + iy \right| = \left| \sqrt{\frac{a + ib}{c + id}} \right|\]

\[ \Rightarrow \left| x + iy \right| = \sqrt{\frac{\left| a + ib \right|}{\left| c + id \right|}}\]

\[ \Rightarrow \sqrt{x^2 + y^2} = \sqrt{\frac{\sqrt{a^2 + b^2}}{\sqrt{c^2 + d^2}}} \left[ \because \left| x + iy \right| = \sqrt{x^2 + y^2} \right]\]

\[\text { Squaring both the sides }, \]

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

\[\text { Squaring again, we get }, \]

\[ \left( x^2 + y^2 \right)^2 = \frac{a^2 + b^2}{c^2 + d^2}\]