Answer
Center: $\left(\frac{5}{2}, 2\right)$
Radius = \frac{3}{2}$ units
$equation: $\left(y-\frac{5}{2}\right)^2+(x-2)^2=\frac{9}{4}$
Work Step by Step
The center $(h,k)$ will be the midpoint of the endpoints of the diameter:
$$(h,k)=\left(\frac{4+1}{2}, \frac{2+2}{2}\right)=\left(\frac{5}{2}, 2\right)$$
The raduis $r$ is equal to one-half of the diameter.
The diameter of the cirlce is equal to the distance between the endpoints $(1, 2)$ and $(4, 2)$.
Since the endpoints share the same $y$-coordinate, the distance between the two points is equal to the difference between their $x$-coordinates:
$\text{diameter} = 4-1=3$
Thus, the radius is $\frac{3}{2}$.
Therefore, the standard form of the equation is
$$\left(y-\frac{5}{2}\right)^2+\left(x-2\right)^2=\left(\frac{3}{2}\right)^2\\
\left(y-\frac{5}{2}\right)^2+\left(x-2\right)^2=\frac{9}{4}$$
