Question

If $b$ is a mean proportional between $a$ and $c$, prove that $\frac{a^2-b^2+c^2}{a^{-2}-b^{-2}+c^{-2}}=b^4$.

Answer 1

Now b is mean proportional between $a,c$ then $b^2=ac$

$\frac{a^2-b^2+c^2}{\frac{1}{a^2}-\frac{1}{b^2}+\frac{1}{c^2}}$

Putting tht value of $b^2$

$=\frac{a^2-ac+c^2}{\frac{1}{a^2}-\frac{1}{ac}+\frac{1}{c^2}}$

$=a^2{c}^2\left(\frac{a^2-ac+c^2}{c^2-ac+a^2}\right)=a^2{c}^2=b^4$