Find the third proportional to $\dfrac{x}{y} + \dfrac{y}{x} \text{ and } \sqrt{x^2 + y^2}$

Let third proportional be p.

$\therefore \dfrac{\dfrac{x}{y} + \dfrac{y}{x}}{\sqrt{x^2 + y^2}} = \dfrac{\sqrt{x^2 + y^2}}{p} \\[1em] \Rightarrow \Big(\sqrt{x^2 + y^2}\Big)^2 = p\Big(\dfrac{x}{y} + \dfrac{y}{x}\Big) \\[1em] \Rightarrow x^2 + y^2 = p \times \dfrac{x^2 + y^2}{xy} \\[1em] \Rightarrow p = \dfrac{(x^2 + y^2)(xy)}{x^2 + y^2} \\[1em] \Rightarrow p = xy.$

Hence, third proportional to $\dfrac{x}{y} + \dfrac{y}{x} \text{ and } \sqrt{x^2 + y^2}$ is xy.