\[\begin{array}{l}
A\left( {a;{a^2}} \right),{\text{ }}B\left( {b;\sqrt {{b^2} - 5} } \right) \hfill \\
\min \left| {AB} \right| = ? \hfill \\
\end{array}\]
\[\begin{array}{l}
f\left( {a,b} \right) = {\left| {AB} \right|^2} = {\left( {a - b} \right)^2} + {\left( {{a^2} - \sqrt {{b^2} - 5} } \right)^2} \hfill \\
\left\{ \begin{array}{l}
{{f'}_a} = 0 \hfill \\
{{f'}_b} = 0 \hfill \\
{\left( {{a^2}} \right)^\prime }_a = {\left( {\sqrt {{b^2} - 5} } \right)^\prime }_b \hfill \\
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
a = \frac{{\sqrt 6 }}{2} \hfill \\
b = \sqrt 6 \hfill \\
\end{array} \right. \hfill \\
\end{array}\]
\[\frac{{\sqrt 7 }}{2}\]