First slide

Algebra of complex numbers

Question

If $x + i y = \sqrt{\frac{a + i b}{c + i d}},$ then ${(x^{2} + y^{2})}^{2} (c^{2} + d^{2})$ equal

Moderate

A

a² + b²
B

$\sqrt{a^{2} + b^{2}}$
C

$\frac{a^{2} + b^{2}}{c^{2} + d^{2}}$
D

$\sqrt{\frac{a^{2} + b^{2}}{c^{2} + d^{2}}}$

Solution

$\begin{array}{l} (x + i y)^{2} (c + i d) = a + i b \\ \Rightarrow |(x + i y)^{2}| | c + i d | = | a + i b | \\ \Rightarrow | x + i y |^{2} | c + i d | = | a + i b | \\ \Rightarrow (x^{2} + y^{2}) \sqrt{c^{2} + d^{2}} = \sqrt{a^{2} + b^{2}} \end{array}$

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