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If \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\), then the value of x2 + y2 is -
1. \(\sqrt {\frac{{{a^2} + {b^2}}}{{{c^2} + {d^2}}}}\)
2. \({\frac{{{a^2} + {b^2}}}{{{c^2} + {d^2}}}}\)
3. ad - bc
4. \(\sqrt {\frac{{a - ib}}{{a + ib}}}\)

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Correct Answer - Option 1 : \(\sqrt {\frac{{{a^2} + {b^2}}}{{{c^2} + {d^2}}}}\)

Calculation:

Given, \(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\)

Taking modulus both sides:

\(\left| {x + ij} \right| = {\left| {\frac{{a + ib}}{{c + id}}} \right|^{\frac{1}{2}}}\)

\(\Rightarrow \sqrt {{x^2} + {y^2}} = {\left( {\frac{{\sqrt {{a^2} + {b^2}} }}{{\sqrt {{c^2} + {d^2}} }}} \right)^{\frac{1}{2}}}\)

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

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