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Question Number 89991 by jagoll last updated on 20/Apr/20

$$\mathrm{if}\mathrm{a},\mathrm{b}>0\mathrm{and}\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}=\frac{5}{3}$$$$\mathrm{find}\frac{{\mathrm{a}}^2+{\mathrm{b}}^2}{\mathrm{ab}}$$

Commented by john santu last updated on 20/Apr/20

$$ab=\sqrt{a^2{b}^2},sincea,b>0$$$$a^2=\frac{5}{3}{b}^2\Rightarrow ab=\sqrt{\frac{5b^4}{3}}=\frac{b^2\sqrt{15}}{3}$$$$then\frac{a^2+b^2}{ab}=\frac{\frac{8b^2}{3}}{\frac{b^2\sqrt{15}}{3}}$$$$=\frac{8}{\sqrt{15}}$$

Commented by mr W last updated on 20/Apr/20

$${\left(\frac{a}{b}\right)}^2=\frac{5}{3}$$$$\frac{a}{b}=\frac{\sqrt{5}}{\sqrt{3}}$$$$\frac{a^2+b^2}{ab}=\frac{a}{b}+\frac{b}{a}=\frac{\sqrt{5}}{\sqrt{3}}+\frac{\sqrt{3}}{\sqrt{5}}=\frac{8}{\sqrt{15}}=\frac{8\sqrt{15}}{15}$$

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