Question

The square root of $4ab-2i(a^2-b^2)$:

• A. $\pm [(a+b)-i(a-b)]$
• B. $\pm [(a+b)+i(a-b)]$
• C. $\pm [\left(ab\right)+i(a+b)]$
• D. $\pm [\left(ab\right)-i(a+b)]$

Answer 1

Answer: (B)

$\pm [(a+b)+i(a-b)]$

Let $(x+iy{)}^2=4ab-2i(a^2-b^2)$

Comparing the real and imaginary terms,

$x^2-y^2=4ab$----------------------------------------(i)

$xy=a^2-b^2$---------------------------------------(ii)

Comparing the magnitude,

$x^2+y^2=\sqrt{4a^4+4b^4+8a^2{b}^2}$

$\Rightarrow x^2+y^2=2a^2+2b^2$ -(iii)

Solving (i) and (iii),

$x^2=a^2+b^2+2ab$ and $y^2=a^2+b^2-2ab$

Hence,

$x=a+b$ , $y=a-b$

or $x=-(a+b)$ , $y=b-a$

Hence, option B is correct