$\begin{aligned} If x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} and \\ y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} then \\ (x + y) equals ? \end{aligned}$ $\eqalign{ & {\text{If }}x = \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }}{\text{ and}} \cr & y = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }}{\text{ then}} \cr & \left( {x + y} \right){\text{equals ?}} \cr}$

18
216
32√15
42(√5 + √3)

Answer:- 1

Explanation:-

Solution:

\begin{aligned} x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \\ \Rightarrow x = \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} \times \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} \\ \Rightarrow x = \frac{{(\sqrt{5} + \sqrt{3})}^{2}}{2} \\ Similarly \\ y = \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} \\ \Rightarrow y = \frac{{(\sqrt{5} - \sqrt{3})}^{2}}{2} \\ Now, x + y \\ = \frac{5 + 3 + 2 \sqrt{15} + 5 + 3 - 2 \sqrt{15}}{2} \\ = \frac{16}{2} \\ = 8 \end{aligned}

$\eqalign{ & x = \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }} \cr & \Rightarrow x = \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }} \times \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }} \cr & \Rightarrow x = \frac{{{{\left( {\sqrt 5 + \sqrt 3 } \right)}^2}}}{2} \cr & {\text{Similarly}} \cr & y = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }} \cr & \Rightarrow y = \frac{{{{\left( {\sqrt 5 - \sqrt 3 } \right)}^2}}}{2} \cr & {\text{Now, }}x + y \cr & = \frac{{5 + 3 + 2\sqrt {15} + 5 + 3 - 2\sqrt {15} }}{2} \cr & = \frac{{16}}{2} \cr & = 8 \cr}$