设直线 $l$ 的参数方程为 $\begin{cases}
x=a+t\cos\alpha\\
y=t\sin\alpha\\
\end{cases}$（$t$ 是参数，$\alpha$ 是倾斜角且 $\alpha\in(0,\pi)$），代入抛物线方程得 $t^2\sin^2\alpha-4t\cos\alpha-4a=0$．设该方程的两根为 $t_1,t_2$，则 $t_1+t_2=\dfrac{4\cos\alpha}{\sin^2\alpha},t_1\cdot t_2=-\dfrac{4a}{\sin^2\alpha}$，则 $\dfrac{1}{|PM{{|}^{2}}}+\dfrac{1}{|QM{{|}^{2}}}=\dfrac{1}{t_{1}^{2}}+\dfrac{1}{t_{2}^{2}}=\dfrac{t_{1}^{2}+t_{2}^{2}}{{{\left({{t}_{1}}{{t}_{2}} \right)}^{2}}}=\dfrac{{{\left( {{t}_{1}}+{{t}_{2}}\right)}^{2}}-2{{t}_{1}}{{t}_{2}}}{{{\left( {{t}_{1}}{{t}_{2}} \right)}^{2}}}=\dfrac{\dfrac{16{{\cos}^{2}}\alpha }{{{\sin }^{4}}\alpha }+\dfrac{8a}{{{\sin }^{2}}\alpha }}{\dfrac{16{{a}^{2}}}{{{\sin}^{4}}\alpha }}=\dfrac{2{{\cos }^{2}}\alpha +a{{\sin }^{2}}\alpha}{2{{a}^{2}}}$ 为常数，所以 $a=2$．