Question

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

Answer 1

$x=\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}$

$y=\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}$

$x^2+xy+y^2$

$={\left(\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\right)}^2+\frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}\cdot \frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}+{\left(\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}+\sqrt{b}}\right)}^2$

$=\frac{{(\sqrt{a}+\sqrt{b})}^2}{{(\sqrt{a}-\sqrt{b})}^2}+1+\frac{{(\sqrt{a}-\sqrt{b})}^2}{{(\sqrt{a}+\sqrt{b})}^2}$

$=\frac{2a^2+12ab+2b^2}{{(\sqrt{a}-\sqrt{b})}^2{(\sqrt{a}+\sqrt{b})}^2}+1$

$=\frac{2a^2+12ab+2b^2}{{(a-b)}^2}+1$

$=\frac{2a^2+12ab+2b^2}{a^2+2ab+b^2}+1$