The equation of a circle in standard form:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
(h, k) - center
r - radius
The center is a midpoint of MN. The formula of a midpoint:
[tex]\left(\dfrac{x_1+x_2}{2};\ \dfrac{y_1+y_2}{2}\right)[/tex]
We have M(2, 4) and N(9, 4). Substitute:
[tex]x=\dfrac{2+9}{2}=\dfrac{11}{2}=5.5\\\\y=\dfrac{4+4}{2}=\dfrac{8}{2}=4[/tex]
Therefore we have the center (5.5, 4) → h = 5.5 and k = 4.
The radius of a distance between the center and M.
The formula of a distance between two points:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
We have the points (5.5, 4) and (2, 4) Substitute:
[tex]d=\sqrt{(4-4)^2+(2-5.5)^2}=\sqrt{0^2+(-3.5)^2}=\sqrt{3.5^2}=3.5[/tex]
Substitute to the equation of a circle:
[tex](x-5.5)^2+(y-4)^2=3.5^2\\\\\boxed{(x-5.5)^2+(y-4)^2=12.25}\to\boxed{C.}[/tex]
