Answer
$\frac{(x-1)^2}{1}+\frac{(y-2)^2}{5}=1$
See graph.
Work Step by Step
1. Given center $(1,2)$, focus $(1,4)$ and contain point $(2,2)$, we can identify a vertical major axis, $c=4-2=2$, $a^2=b^2+c^2=b^2+4$
2. Write the equation as $\frac{(x-1)^2}{b^2}+\frac{(y-2)^2}{a^2}=1$, use the point on curve to get $\frac{(2-1)^2}{b^2}+\frac{(2-2)^2}{a^2}=1$, thus $b=1$ and $a^2=5$
3. Thus we have $\frac{(x-1)^2}{1}+\frac{(y-2)^2}{5}=1$
4. See graph.
