Question

If $a, b, c$ and $d \in R$ such that $a^{2}+b^{2}=4$ and $c^{2}+d^{2}=2$ and if $(a+i b)^{2}=(c+i d)^{2}(x+i y)$, then $x^{2}+y^{2}$ is equal to

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

Answer: (A)

Given, $(a+i b)^{2}=(c+i d)^{2}(x+i y)$
$\Rightarrow |a+i b|^{2}=|c+i d|^{2}|x+i y|$
$\Rightarrow a^{2}+b^{2}=\left(c^{2}+d^{2}\right)\left(\sqrt{x^{2}+y^{2}}\right)$
$\Rightarrow 4= 2\left(\sqrt{x^{2}+y^{2}}\right)$
($\because a^{2}+b^{2}=4$ and $c^{2}+d^{2}=2$ given )
$ \Rightarrow \sqrt{x^{2}+y^{2}}=2 $
$ \Rightarrow x^{2}+y^{2}=4 $