Key Concepts and Formulas

• Equation of a Circle: The standard equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$.
• Circle Tangent to x-axis: If a circle touches the x-axis at the point $(a, 0)$, then its center must have coordinates $(a, \pm r)$ where $r$ is the radius.

Step-by-Step Solution

Step 1: Determine the Center of the Circle

• Explanation: We're given the circle touches the x-axis at $(1, 0)$. This significantly constrains the possible center locations.
• Reasoning: Since the circle is tangent to the x-axis at $(1, 0)$, the center must lie on the vertical line $x = 1$. Let the center be $(1, k)$. The radius of the circle is therefore $|k|$. Since the circle also passes through $(2, 3)$, and $3 > 0$, the center must be above the x-axis, meaning $k > 0$. Therefore, the radius is simply $k$. Let $r = k$. So, the center is $(1, r)$.

Step 2: Apply the Distance Formula

• Explanation: We know the circle passes through $(2, 3)$, and we know the center is $(1, r)$. The distance between these two points must equal the radius $r$.
• Reasoning: Using the distance formula: $r = \sqrt{(2-1)^2 + (3-r)^2}$ Squaring both sides to eliminate the square root: $r^2 = (2-1)^2 + (3-r)^2$

Step 3: Solve for the Radius

• Explanation: We now have an equation with only one unknown, $r$. We solve for it.
• Reasoning: $r^2 = 1 + (9 - 6r + r^2)$ $r^2 = 10 - 6r + r^2$ Subtracting $r^2$ from both sides: $0 = 10 - 6r$ $6r = 10$ $r = \frac{10}{6} = \frac{5}{3}$

Step 4: Calculate the Diameter

• Explanation: The problem asks for the diameter, which is twice the radius.
• Reasoning: Diameter $= 2r = 2 \times \frac{5}{3} = \frac{10}{3}$

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs when applying the distance formula and expanding squared terms.
• Forgetting the Diameter: Double-check that you are answering the question asked. It's easy to solve for the radius and forget to calculate the diameter.
• Visualizing the Problem: Drawing a quick sketch can help you understand the geometry and avoid errors in setting up the equations.

Summary

By using the fact that the circle is tangent to the x-axis at (1, 0), we deduced that the center must be at (1, r). Then, using the distance formula with the point (2, 3), we solved for the radius r = 5/3. Finally, we calculated the diameter as twice the radius, which gives us 10/3.

The final answer is \boxed{\frac{10}{3}}, which corresponds to option (A).