Example 4. Given point $\mathrm{A}$ as a fixed point on a plane, and $\mathrm{F}$ as the focus of a parabola, try to find a point $\mathrm{P}$ on the parabola such that $|P A|+|P F|$ is minimized.
Solve as shown in the figure
5, discuss in two cases: (1)
If $\mathrm{AF}$ intersects the parabola
at $P_{1}$, then
$\mathrm{P}_{1} \mathrm{~A} + \mathrm{P}_{1} \mathrm{~F}$
$=\mathrm{AF}$, i.e., $\mathrm{P}_{1}$ is
the solution. (2) If $\mathrm{AF}$
does not intersect the parabola,
then draw $\mathrm{AB} \perp \mathrm{L}$, ( $\mathrm{B}$ is the foot of the perpendicular), $\mathrm{AB}$ intersects
the parabola at $\mathrm{P}_{2}, \because \mathrm{P}_{2} \mathrm{~A} + \mathrm{P}_{2} \mathrm{~F} = \mathrm{P}_{2} \mathrm{~A}$
$+\mathrm{P}_{2} \mathrm{~B}=\mathrm{AB}$, i.e., $\mathrm{P}_{2}$ is the solution.