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Step-by-Step Solution
Step 1: Express the center and radius of the circle
Let the center of the circle be at the point $(r,\, 6)$ and its radius be $r$.
Since the circle touches the y-axis at $(0,\,6)$, the center must be horizontally
$r$ units away from the y-axis. Therefore, the circle's center can be written as
$(r,\, 6)$, and the distance from this center to the line $x=0$ (the y-axis) is
precisely $r$.
Step 2: Write the equation of the circle
The general equation of a circle with center $(h,\, k)$ and radius $R$ is
$$(x-h)^2 + (y-k)^2 = R^2.$$
In this problem, $h = r$, $k = 6$, and $R = r$. So the equation becomes:
$$
(x - r)^2 + (y - 6)^2 = r^2.
$$
Step 3: Find the x-intercept of the circle
To find points where the circle meets the x-axis, set $y = 0$ in the circle's equation:
$$
(x - r)^2 + (0 - 6)^2 = r^2.
$$
Simplifying,
$$
(x - r)^2 + 36 = r^2
\quad \Longrightarrow \quad
(x - r)^2 = r^2 - 36.
$$
Hence,
$$
x - r = \pm \sqrt{r^2 - 36}.
$$
Therefore, the two x-intercepts are:
$$
x_1 = r - \sqrt{r^2 - 36},
\quad
x_2 = r + \sqrt{r^2 - 36}.
$$
Step 4: Use the given length of the x-intercept
We are told the circle cuts off an intercept of $6\sqrt{5}$ on the x-axis.
The distance between the two x-intercepts is:
$$
x_2 - x_1
= \bigl(r + \sqrt{r^2 - 36}\bigr) - \bigl(r - \sqrt{r^2 - 36}\bigr)
= 2\sqrt{r^2 - 36}.
$$
According to the problem:
$$
2\sqrt{r^2 - 36} = 6\sqrt{5}.
$$
Divide both sides by 2:
$$
\sqrt{r^2 - 36} = 3\sqrt{5}.
$$
Square both sides:
$$
r^2 - 36 = 45
\quad \Longrightarrow \quad
r^2 = 81
\quad \Longrightarrow \quad
r = 9 \quad (\text{taking the positive value for radius}).
$$
Step 5: Conclude the radius
Thus, the radius of the circle is $\boxed{9}$.
