Let a curve $y=f(x), x \in(0, \infty)$ pass through the points $P\left(1, \frac{3}{2}\right)$ and $Q\left(a, \frac{1}{2}\right)$. If the tangent at any point $R(b, f(b))$ to the given curve cuts the $\mathrm{y}$-axis at the point $S(0, c)$ such that $b c=3$, then $(P Q)^{2}$ is equal to __________.