圆心已经给出，所以只需要用点 $B$ 求半径。圆的标准形式是 $(x-a)^2+(y-b)^2=r^2$。

The centre is $A(3,5)$.

The radius is $AB$, so

Therefore, an equation of $C$ is

水平弦的中点在圆心正上方或正下方。因为弦在 $x$-axis 上方，所以选圆心上方。半弦长是 $2\sqrt{22}$，半径是 $13$，用直角三角形求圆心到弦的距离。

The radius of the circle is

Since $MN$ has length $4\sqrt{22}$, half of the chord has length

Let the perpendicular distance from $A$ to the chord $MN$ be $d$.

Using Pythagoras’ theorem,

The centre is $A(3,5)$, and $MN$ lies above the $x$-axis, so the $y$-coordinate of $MN$ is

Therefore, the equation of the line through $M$ and $N$ is

由于 $MN$ 平行于 $x$ 轴，设直线方程为 $y=k$（且 $k>0$）。因为 $MN$ 的长度为 $4\sqrt{22}$，且圆的对称轴为 $x=3$，所以 $M, N$ 的 $x$ 坐标为 $3 \pm 2\sqrt{22}$。将其代入圆方程中可以直接求解 $k$。

Since the chord $MN$ is parallel to the $x$-axis, its equation can be written as $y = k$ (where $k > 0$).

The centre of the circle is $A(3,5)$ and the length of $MN$ is $4\sqrt{22}$, which means the $x$-coordinates of the points $M$ and $N$ are symmetrically placed at:

Substitute these coordinates into the equation of $C$ from part (a):

This gives two possible values:

Since the chord lies above the $x$-axis, we must have $k > 0$, so $k = 14$.

Therefore, the equation of the line is $y = 14$.