Mathematics Question from JEE Main 2023 · 6 April, Shift 2

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