Question

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

Answer 1

Let the required third proportional be p.

${\displaystyle \Rightarrow \frac{x}{y}+\frac{y}{x},\sqrt{x^2+y^2}}$, p are in continued proportion

${\displaystyle \Rightarrow \frac{x}{y}+\frac{y}{x}:\sqrt{x^2+y^2}=\sqrt{x^2+y^2}:p}$

${\displaystyle \Rightarrow p(\frac{x}{y}+\frac{y}{x})={\left(\sqrt{x^2+y^2}\right)}^2}$

${\displaystyle \Rightarrow p\left(\frac{x^2+y^2}{xy}\right)=x^2+y^2}$

${\displaystyle \Rightarrow }$ p = xy