Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Prove that the graph of $y = f ( x )$ and $\Omega$ have a common tangent line at point $P$.