Question

If $\sqrt{\left(x+iy\right)}=\pm \left(a+ib\right)$, then $\sqrt{\left(-x-iy\right)}$ is equal to

• A. $\pm \left(b+ia\right)$
• B. $\pm \left(a+ib\right)$
• C. $\pm \left(a-ib\right)$
• D. $\pm \left(b-ia\right)$
• E. None of these

Answer 1

The correct option is D

$\pm \left(b-ia\right)$

Explanation for the correct option:

Step 1:Solve the given relation by squaring both sides,

Given relation $\sqrt{\left(x+iy\right)}=\pm \left(a+ib\right)$

$\begin{array}{rcl}\left(x+iy\right)& =& {\left(a+ib\right)}^2\\ & =& a^2-b^2+i2ab\end{array}$

Step 2: Compare the real and imaginary parts, obtain

$\begin{array}{rcl}x& =& a^2-b^2\\ y& =& 2ab\end{array}$

From above

$\begin{array}{rcl}\left(-x-iy\right)& =& -\left(a^2-b^2\right)-2iab\\ & =& -a^2+b^2-2iab\\ & =& b^2-a^2-2iab\\ & =& {\left(b\right)}^2+{\left(ia\right)}^2-2ia\left(b\right)\\ & =& {\left(b-ia\right)}^2\end{array}$

Take square roots on both the sides of the above equation,

Hence, the correct option is (E).