If $x + i y = \sqrt{(a + i b) / (c + i d)}$ , then $(x^{2} + y^{2})^{2} = (a^{2} + b^{2}) / (c^{2} + d^{2})$ . If this is true enter 1, else enter 0.

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Solution

$x + i y = \sqrt{(a + i b) / (c + i d)}$
by taking modulus
$| x + i y | = \sqrt{∣ ∣ ∣ \frac{a + i b}{c + i d} ∣ ∣ ∣}$
$\sqrt{x^{2} + y^{2}} = \sqrt{\sqrt{\frac{a^{2} + b^{2}}{c^{2} + d^{2}}}}$
$\Rightarrow x^{2} + y^{2} = \sqrt{\frac{a^{2} + b^{2}}{c^{2} + d^{2}}}$
$\Rightarrow {(x^{2} + y^{2})}^{2} = \frac{a^{2} + b^{2}}{c^{2} + d^{2}}$
Ans: 1

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x - i y = \sqrt{\frac{a - i b}{c - i d}}

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