Question

Find the third proportional to $$\left(\frac{a}{b} +\frac{b}{a}\right)$$ and $$\sqrt{a^2 + b^2}$$.

Answer 1

If $$x$$ is the third proportional to $$a$$ and $$b$$, then $$a : b :: b : x \Rightarrow x = \frac{b^2}{a}$$.

$$a = \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab}$$ and $$b = \sqrt{a^2 + b^2}$$

$$x = \frac{(\sqrt{a^2 + b^2})^2}{\frac{a^2 + b^2}{ab}}$$

$$= \frac{(a^2 + b^2) \times ab}{a^2 + b^2}$$

$$= ab$$