Question

If $\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=\frac{10}{3}$, Then find $\frac{x}{y}+\frac{y}{x}$

Answer 1

$\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=\frac{10}{3}$

squaring both sides, we get

${(\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}})}^2=\frac{100}{9}$

$\Rightarrow \frac{x}{y}+\frac{y}{x}+2\frac{x}{y}\times \frac{y}{x}=\frac{100}{9}$

$\Rightarrow \frac{x}{y}+\frac{y}{x}+2=\frac{100}{9}$

$\Rightarrow \frac{x}{y}+\frac{y}{x}=\frac{100}{9}-2=\frac{100-18}{2}=\frac{82}{2}=41$

$\therefore \frac{x}{y}+\frac{y}{x}=41$