Key Concepts and Formulas

• Equation of a Circle: A circle with center $(h, k)$ and radius $r$ has the equation $(x - h)^2 + (y - k)^2 = r^2$.
• Tangent to an Axis: If a circle is tangent to the y-axis at $(0, a)$, the center is $(\pm r, a)$, where $r$ is the radius. Similarly, if tangent to the x-axis at $(b, 0)$, the center is $(b, \pm r)$.
• Chord Bisector Theorem: A perpendicular line from the center of a circle to a chord bisects the chord. This creates a right triangle where the radius is the hypotenuse.

Step-by-Step Solution

Step 1: Define the circle's equation based on the tangency condition.

• What we're doing: We use the given information that the circle touches the y-axis at (0, 6) to deduce the center's coordinates in terms of the radius and then define the circle's equation.
• Why: This allows us to represent the circle algebraically and incorporate the given geometric constraint.
• Mathematical Representation: Since the circle touches the y-axis at (0, 6), the center of the circle must be at $(r, 6)$ or $(-r, 6)$, where $r$ is the radius. We can assume the center is at $(r, 6)$ without loss of generality, as squaring $r$ later eliminates the sign ambiguity. The equation of the circle is then $(x - r)^2 + (y - 6)^2 = r^2$.

Step 2: Use the x-intercept information and the chord bisector theorem.

• What we're doing: We use the information about the x-intercept to relate the radius to the distance from the center to the x-axis using the chord bisector theorem.
• Why: This establishes a relationship that allows us to solve for the radius.
• Mathematical Representation: The circle cuts off an intercept of length $6\sqrt{5}$ on the x-axis. Consider the perpendicular from the center $(r, 6)$ to the x-axis. This perpendicular has length 6. It bisects the chord of length $6\sqrt{5}$, so half the chord length is $3\sqrt{5}$. By the Pythagorean theorem: $r^2 = 6^2 + (3\sqrt{5})^2$

Step 3: Solve for the radius.

• What we're doing: We solve the equation derived in the previous step for the radius, $r$.
• Why: This gives us the numerical value of the radius.
• Mathematical Representation: $r^2 = 36 + (9 \times 5)$ $r^2 = 36 + 45$ $r^2 = 81$ $r = \sqrt{81} = 9$ Since the radius must be positive, $r = 9$.

Common Mistakes & Tips

• Sign of the x-coordinate of the center: Remember that the center could be at $(r, 6)$ or $(-r, 6)$. Choosing $(r, 6)$ simplifies the calculations, but the final result for $r^2$ will be the same.
• Pythagorean Theorem: Ensure you correctly identify the hypotenuse (radius) and the legs of the right triangle.
• Squaring terms: Be careful when squaring terms like $3\sqrt{5}$.

Summary

We used the tangency condition to express the circle's equation in terms of its radius. Then, we applied the chord bisector theorem and the Pythagorean theorem to relate the radius to the x-intercept length. Solving for the radius, we find that the radius of the circle is 9.

The final answer is $\boxed{9}$, which corresponds to option (B).