root(5+12i) + root(5-12i)/ root(5+12i) - root(5-12i) =

1. -3/2 i 2. 3/2i

3. -3/2 4. 3/2

the given expression is: $\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}$
to rationalise the denominator, multiply the numerator and denominator by $\sqrt{5+12i}+\sqrt{5-12i}$ .
$$\begin{gathered} \frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}.\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}+\sqrt{5-12i}} \\ =\frac{{\left[\sqrt{5+12i}+\sqrt{5-12i}\right]}^2}{(5+12i)-(5-12i)} \\ =\frac{5+12i+5-12i+2.\sqrt{5+12i}.\sqrt{5-12i}}{5+12i-5+12i} \\ =\frac{10+2\sqrt{(5+12i)(5-12i)}}{24i} \\ =\frac{10+2\sqrt{25-144i^2}}{24i} \\ =\frac{10+2\sqrt{25+144}}{24i} \end{gathered}$$
$$\begin{gathered} =\frac{10+2.13}{24i} [\sqrt{25+144}=\sqrt{169}=13 \\ =\frac{10+26}{24i} \\ =\frac{36}{24i}=\frac{3}{2i} \end{gathered}$$

thus option (2) is correct.

hope this helps you