Question

Find the third proportional to $\frac{x}{y}+\frac{y}{x}$ and $\frac{x}{y}$.

Answer 1

For two number $a,b$ the third proportional $c$ is such tha $\frac{a}{b}=\frac{b}{c}$

$c=\frac{b^2}{a}$

So in the question the third proportional is

$=\frac{\frac{x^2}{y^2}}{\frac{x}{y}+\frac{y}{x}}=\frac{\frac{x^2}{y^2}}{\frac{x^2+y^2}{xy}}=\frac{x^2}{y^2}.\frac{xy}{x^2+y^2}=\frac{x^3}{y(x^2+y^2)}$