Prove that the arithmetic mean of the roots of $x^{2} - 2 a x + b^{2} = 0$ is equal to the geometric mean of the roots of the equation $x^{2} - 2 b x + a^{2} = 0$ .

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Solution

Let the roots of equation $x^{2} - 2 a x + b^{2}$ be $m$ and $n$
Let the roots of equation $x^{2} - 2 b x + a^{2}$ be $p$ and $q$
$A M$ of $m$ and $n$ $= \frac{m + n}{2} = \frac{2 a}{2} = a$
$G M$ of $p$ and $q$ $= \sqrt{p q} = \sqrt{a^{2}} = a$

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