Question

$Im\left(\sqrt{a+i\sqrt{a^4+a^2+1}}\right)=$

• A. $\sqrt{\frac{a^2+a+1}{2}}$
• B. $\sqrt{\frac{a^2-a+1}{2}}$
• C. $\frac{1}{2}\sqrt{a^2-a+1}$
• D. $\frac{1}{2}\sqrt{a^2+a+1}$

Answer 1

The correct option is B $\sqrt{\frac{a^2-a+1}{2}}$

$z=x+iy$ (say)
Squaring both sides & comparing the real & imaginary parts
$(x+iy{)}^2=a+i\sqrt{a^4+a^2+1}$

$x^2-y^2+2xyi=a+i\sqrt{a^4+a^2+1}$
$x^2-y^2=a$-------------------------------(1)
$2xy=\sqrt{a^4+a^2+1}$------------------------------------(2)
$(x+iy{)}^2=a+i\sqrt{a^4+a^2+1}$
taking modulus,
$x^2+y^2=\sqrt{a^2+a^4+a^2+1}=\sqrt{(a^2{)}^2+2(a^2)1+1}$
$=\sqrt{(a^2+1{)}^2}$
$=a^2+1$
$\therefore x^2+y^2=a^2+1$-----------------------------(3)

From (1) & (3),

$x^2=\frac{a^2+a+1}{2}$

$\therefore y^2=x^2-a=\frac{a^2+a+1}{2}-a$

$y^2=\frac{a^2-a+1}{2}$

$\therefore y=\sqrt{\frac{a^2-a+1}{2}}$