If q is the mean proportional between p and r, prove that: \[{{p}^{2}}-3{{q}^{2}}+{{r}^{2}}={{q}^{4}}(\frac{1}{{{p}^{2}}}-\frac{3}{{{q}^{2}}}+\frac{1}{{{r}^{2}}})\]

It is given that

q is the mean proportional between p and r

q² = pr

Here

LHS = \[{{p}^{2}}~\text{ }3{{q}^{2}}~+\text{ }{{r}^{2}}\]

We can write it as

\[=\text{ }{{p}^{2}}~\text{ }3pr\text{ }+\text{ }{{r}^{2}}\]

So we get

\[=\text{ }{{r}^{2}}~\text{ }3pr\text{ }+\text{ }{{p}^{2}}\]

Here

LHS = RHS

Therefore, it is proved.