Question

Find the value of $\frac{\sqrt{5+12\mathfrak{i}}+\sqrt{5-12\mathfrak{i}}}{\sqrt{5+12\mathfrak{i}}-\sqrt{5-12\mathfrak{i}}}$?

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

Answer 1

The correct option is A

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

Explanation for correct option

Finding the value of the given expression

Given,

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

Multiplying and dividing by $\sqrt{5+12\mathfrak{i}}+\sqrt{5-12\mathfrak{i}}$:

$$\begin{gathered} =\frac{{\left(\sqrt{5+12\mathfrak{i}}+\sqrt{5-12\mathfrak{i}}\right)}^2}{{\left(\sqrt{5+12\mathfrak{i}}\right)}^2-{\left(\sqrt{5-12\mathfrak{i}}\right)}^2} \\ =\frac{{\left(\sqrt{5+12\mathfrak{i}}\right)}^2+{\left(\sqrt{5-12\mathfrak{i}}\right)}^2+2\left(\sqrt{5+12\mathfrak{i}}\right)\left(\sqrt{5-12\mathfrak{i}}\right)}{\left(5+12\mathfrak{i}\right)-\left(5-12\mathfrak{i}\right)} \\ =\frac{5+12\mathfrak{i}+5-12\mathfrak{i}+2\left(\sqrt{{\left(5\right)}^2-{\left(12\mathfrak{i}\right)}^2}\right)}{24\mathfrak{i}} \\ =\frac{10+2\left(\sqrt{25-\left(-144\right)}\right)}{24\mathfrak{i}} \\ =\frac{10+2\sqrt{169}}{24\mathfrak{i}} \\ =\frac{10+2\times 13}{24\mathfrak{i}} \\ =\frac{10+26}{24\mathfrak{i}} \\ =\frac{36}{24\mathfrak{i}} \\ =\frac{3\mathfrak{i}}{2{\left(\mathfrak{i}\right)}^2}\left(\mathrm{Multiply}\mathrm{and}\mathrm{divide}\mathrm{by}\mathfrak{i}\right) \\ =\frac{-3\mathfrak{i}}{2}\left(\because {\left(\mathfrak{i}\right)}^2=-1\right) \end{gathered}$$

Hence, the correct answer is Option (A)