Discussion
The A.M. and G.M. of two positive real numbers \(a\) and \(b\) \((a \gt b)\) are \(A\) and \(G\) respectively. If \(A : G = 5 : 3\), then \(a^{2} + b^{2} : ab =\)
Answer: B
The ratio is 82:9.
Take \(A = 5k\) and \(G = 3k\). Then \(a + b = 2A = 10k\) and \(ab = G^{2} = 9k^{2}\).
\[ \begin{aligned} a^{2} + b^{2} &= (a + b)^{2} - 2ab \\ &= 100k^{2} - 18k^{2} = 82k^{2} \\ a^{2} + b^{2} : ab &= 82 : 9 \end{aligned} \]
The option 83:9 is a one-off slip in the subtraction. The condition \(a \gt b\) only fixes which root is which; it does not change \(a^{2} + b^{2}\) or \(ab\).
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