Q. If $x + iy = \sqrt{\frac{a + ib}{c + id}}$, then $x^{2} + y^{2}$ is equal to
A) $\frac{a^{2} - b^{2}}{c^{2} + d^{2}}$
B) $\frac{a^{2} + b^{2}}{c^{2} + d^{2}}$
C) $\frac{a^{2} + b^{2}}{c^{2} - d^{2}}$
D) None of these
Explanation
(d)
Given, $x + iy = \left( \frac{a + ib}{c + id} \right)^{1/2}$
$\Rightarrow \ \ \ |x + iy| = \left| \frac{a + ib}{c + id} \right|^{1/2}$
(Taking modulus from both side and using $\left| z^{n} \right| = |z|^{n}$)
$\Rightarrow \ \ \ \ |x + iy|^{2} = \left| \frac{a + ib}{c + id} \right|$
$\Rightarrow \ \ \ x^{2} + y^{2} = \sqrt{\frac{a^{2} + b^{2}}{c^{2} + d^{2}}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.