Question-34

algebra
aptitude

For positive real variables \(p\) and \(q\), if

\[ \log(p^{2}+q^{2})=\log p+\log q+2 \log 3 \]

then, the value of \(\cfrac{p^{4}+q^{4}}{p^{2}q^{2}}\) is

We have:

\[ \begin{aligned} \log\left( p^{2} +q^{2}\right) & =\log p+\log q+2\log 3\\ & \\ \log\left(\frac{p^{2} +q^{2}}{pq}\right) & =\log 9 \end{aligned} \]

Multiplying both sides by \(\displaystyle 2\):

\[ \begin{aligned} \log\left(\frac{p^{4} +q^{4} +2p^{2} q^{2}}{p^{2} q^{2}}\right) & =\log 81 \end{aligned} \]

Since \(\displaystyle \log\)is a one-one function, we have:

\[ \begin{aligned} \frac{p^{4} +q^{4}}{p^{2} q^{2}} +2 & =81\\ & \\ \frac{p^{4} +q^{4}}{p^{2} q^{2}} & =79 \end{aligned} \]