The HM and A.M of two positive numbers are in the ratio 4:9. Then the numbers are in the ratio

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Professor PC Thomas Classes · Answer

$$ \begin{align*} & 	ext { } \frac{\frac{2 a b}{a+b}}{\frac{a+b}{2}}=\frac{4}{9} 	ext { ie } \frac{4 a b}{(a+b)^2}=\frac{4}{9} \quad-(1) \ & \Rightarrow \frac{4 a b}{(a+b)^2-4 a b}=\frac{4}{5} 	ext { ie } \frac{4 a b}{(a-b)^2}=\frac{4}{5}  	ag{1}\ & 	ext { (1) } \div(2) \Rightarrow\left(\frac{a-b}{a+b}ight)^2=\frac{5}{9} \Rightarrow \frac{a-b}{a+b}=\frac{\sqrt{5}}{3} 	ext { if } a>b  	ag{2}\ & \Rightarrow \frac{a}{b}=\frac{3+\sqrt{5}}{3-\sqrt{5}} .  \end{align*} $$