Question

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

• A. $-\frac{3}{2}i$
• B. $\frac{3}{2}i$
• C. $-\frac{3}{2}$
• D. $\frac{3}{2}$

Answer 1

Answer: (B)

Given:

$\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}$

Lets move through rationalize the denominator,

$=\left(\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}\right)\times \left(\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}+\sqrt{5-12i}}\right)$

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

$=\left(\frac{(5+12i)+(5-12i)+2\sqrt{5^2-(12i{)}^2}}{5+12i-5+12i}\right)$

$=\frac{10+2\sqrt{25+144}}{24i}$

$=\frac{10+2\cdot 13}{24i}\cdot \frac{i}{i}$

$=\frac{36i}{24i^2}$

$=\frac{3i}{2(-1)}$ [$\because i^2=-1$]

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

Hence, Option A is correct.