Answer
$\frac{(x-1)^2}{9}+\frac{(y-2)^2}{9}=1$
See graph.
Work Step by Step
1. Given center $(1,2)$, vertex $(4,2)$ and contain point $(1,5)$, we can identify a horizontal major axis, $a=4-1=3$, $b^2+c^2=a^2=9$
2. Write the equation as $\frac{(x-1)^2}{a^2}+\frac{(y-2)^2}{b^2}=1$, use the point on curve to get $\frac{(1-1)^2}{a^2}+\frac{(5-2)^2}{b^2}=1$, thus $b=3$ and $c^2=9-9=0$, a circle.
3. Thus we have $\frac{(x-1)^2}{9}+\frac{(y-2)^2}{9}=1$
4. See graph.
