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The ratio of the A.M. and G.M. of two positive numbers a and b

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The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that a : b = $(m + m^{2} - n^{2}) : (m - m^{2} - n^{2})$ .

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The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that a : b =

(m + m^{2} - n^{2}) : (m - m^{2} - n^{2})

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Text solutionVerified

Solution: Here, we are given,

\frac{\frac{a + b}{2}}{a b} = \frac{m}{n}

\Rightarrow \frac{a + b}{2 a b} = \frac{m}{n} \to (1)

Squaring both sides,

\Rightarrow \frac{( a + b ) ^{2}}{4 a b} = \frac{m ^{2}}{n ^{2}}

\Rightarrow \frac{( a + b ) ^{2} - 4 a b}{4 a b} = \frac{m ^{2} - n ^{2}}{n ^{2}}

\Rightarrow \frac{( a - b ) ^{2}}{4 a b} = \frac{m ^{2} - n ^{2}}{n ^{2}}

\Rightarrow \frac{a - b}{2 a b} = \frac{m ^{2} - n ^{2}}{n} \to (2)

Adding (1) and (2),

\frac{2 a}{2 a b} = \frac{m + m ^{2} - n ^{2}}{n} \to (3)

Subtracting (2) from (1),

\frac{2 b}{2 a b} = \frac{m - m ^{2} - n ^{2}}{n} \to (4)

Dividing (3) by (4),

\frac{a}{b} = \frac{m + m ^{2} - n ^{2}}{m - m ^{2} - n ^{2}}

So,

a : b = (m + m^{2} - n^{2}) : (m - m^{2} - n^{2})

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Question Text	The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that a : b = $(m + m^{2} - n^{2}) : (m - m^{2} - n^{2})$ .
Answer Type	Text solution:1
Upvotes	151