Complex Numbers 2 Question 46
47.
If $x+i y=\sqrt{\frac{a+i b}{c+i d}}$, prove that $\left(x^{2}+y^{2}\right)^{2}=\frac{a^{2}+b^{2}}{c^{2}+d^{2}}$
$(1978,2 M)$
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Solution:
- Since, $(x+i y)^{2}=\frac{a+i b}{c+i d}$
$\Rightarrow \quad|x+i y|^{2}=\frac{|a+i b|}{|c+i d|} \quad \because\left|\frac{z _1}{z _2}\right|=\frac{\left|z _1\right|}{\left|z _2\right|}$
$\Rightarrow \quad\left(x^{2}+y^{2}\right)=\frac{\sqrt{a^{2}+b^{2}}}{\sqrt{c^{2}+d^{2}}}$
$\Rightarrow \quad\left(x^{2}+y^{2}\right)^{2}=\frac{a^{2}+b^{2}}{c^{2}+d^{2}}$
Hence proved.
