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The line $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ and $b$ are positive constants, cuts the $x$-axis at $P$ and the $y$-axis at $Q$. Given that $PQ = \sqrt{45}$ and that the gradient of the line $PQ$ is $-\frac{1}{2}$, determine the values of $a$ and $b$.