Answer
Please see the graph.
Work Step by Step
The red graph is $ x^{2}+\left(y-2\right)^{2}\ge9$, and the blue graph is $\frac{x^{2}}{4}+\frac{y^{2}}{25\ }<1$.
The red graph has a greater than or equal to sign, so it will have a solid line. The blue graph has a less than sign, so it will have a dotted line. We also pick the point $(0,2)$ to determine what sides of the graphs to shade.
$(0,2)$
$x^{2}+(y-2)^{2}\ge 9$
$0^{2}+ (2-2)^{2}\ge 9$
$0 + 0^2 \ge 9$
$0 \ge 9$ (false, so we shade the side of the graph without the point)
$(0,2)$
$\frac{x^{2}}{4}+\frac{y^{2}}{25\ }<1$
$\frac{0^{2}}{4}+\frac{2^{2}}{25\ }<1$
$0/4 + 4/25 < 1$
$4/25 < 1$ (true, so we shade the side of the graph with the point)
The overlap of the two graphs is the solution set.
