Question

If $x=\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}},y=\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}$ then the value of $x^2+xy+y^2$ is

• A. $\frac{4(a-b)}{(a+b)}$
• B. $\frac{4(a+b)}{(a-b)}$
• C. $\frac{2(a+b)}{a-b}$
• D. $3\frac{{(a+b)}^2+4ab}{{\left(a-b\right)}^2}$

Answer 1

Answer: (D)

$3\frac{{(a+b)}^2+4ab}{{\left(a-b\right)}^2}$

$\begin{array}{c}Multiply\left(\sqrt{a}+\sqrt{b}\right)inbothnumeratoranddenomiratorinxandmultiply\\ \left(\sqrt{a}-\sqrt{b}\right)inbothnumeratordenomiratorinyweget.\\ x=\frac{a+b+2\sqrt{ab}}{a-b},y=\frac{a+b-2\sqrt{ab}}{a-b}\\ x^2+y^2+xy={\left(\frac{a+b+2\sqrt{ab}}{a-b}\right)}^2\left(\frac{a+b-2\sqrt{ab}}{a-b}\right)+\left(\frac{a+b+2\sqrt{ab}}{a-b}\right)\left(\frac{a+b-2\sqrt{ab}}{a-b}\right)\\ =\frac{{\left(a+b\right)}^2+4ab+4\sqrt{ab}\left(a+b\right)}{{\left(a-b\right)}^2}+\frac{{\left(a+b\right)}^2+4ab2\sqrt{ab}\left(a+b\right)}{{\left(a-b\right)}^2}+\frac{{\left(a+b\right)}^2-4ab}{{(a-b)}^2}\\ =3\frac{{(a+b)}^2+4ab}{{\left(a-b\right)}^2}\\ Thiswillbethewriteanswar\\ \end{array}$