Question

If G is the geometric mean of X and Y, then
1/(G^2 – X^2) + 1/(G^2 – Y^2) =?

Answer 1

Geometric mean of X and Y is \(\sqrt{XY}\), so:
\[\frac{1}{G^2-X^2}+\frac{1}{G^2-Y^2}=\frac{1}{\left(\sqrt{XY}\right)^2-X^2}+\frac{1}{\left(\sqrt{XY}\right)^2-Y^2}\\
=\frac{1}{XY-X^2}+\frac{1}{XY-Y^2}\\
=\frac{1}{X(Y-X)}+\frac{1}{Y(X-Y)}\]

Answer 2

\[= \frac{Y-X}{XY(Y-X)} \]
\[= \frac{1}{XY}\]
\[=\frac{1}{G^2}\]