Question

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

Answer 1

Let the roots of equation $x^2-2ax+b^2$ be $m$ and $n$

Let the roots of equation $x^2-2bx+a^2$ be $p$ and $q$

$AM$ of $m$ and $n$ $=\frac{m+n}{2}=\frac{2a}{2}=a$

$GM$ of $p$ and $q$ $=\sqrt{pq}=\sqrt{a^2}=a$