If P = 7 + 4√3 and PQ = 1, then what is the value of 1P^2 + 1Q^2?

Answer: Option B

$$\begin{array}{rl}& P=7+4\sqrt{3}\\ & PQ=1,Q=\frac{1}{P}\\ & Q=\frac{1}{7+4\sqrt{3}}=7-4\sqrt{3}\\ & P+Q\\ & =7+4\sqrt{3}+7-4\sqrt{3}\\ & =14\\ & \Rightarrow \frac{1}{P^2}+\frac{1}{Q^2}\\ & =\frac{P^2+Q^2}{{\left(PQ\right)}^2}\\ & =\frac{{\left(P+Q\right)}^2-2PQ}{{\left(PQ\right)}^2}\\ & ={\left(14\right)}^2-2\\ & =196-2\\ & =194\end{array}$$