Question

Find the modulus, argument and the principal argument of the complex number.
$z=\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}$.

Answer 1

$z=\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}\times \frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}+\sqrt{5-12i}}$

$=\frac{5+12i+5-12i+2\sqrt{25+144}}{(5+12i)-(5-12i)}$

$=\frac{10+2\left(13\right)}{24i}=\frac{36}{24i}\times \frac{i}{i}=\frac{36i}{-24}=\frac{-3}{2}i$

$z=\frac{-3}{2}i$

Modulus= $\left|\frac{3}{2}\right|=3/2$

Argument= ${\tan }^{-1}\left(\frac{\frac{-3}{2}}{0}\right)=\Pi /2$

Principal Argument= $\Pi /2$