Key Concepts and Formulas

• Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• Distance from a Point to a Line: The distance from point $(x_1, y_1)$ to the line $Ax + By + C = 0$ is $D = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.
• Tangency Condition: If a circle touches a line, the distance from the circle's center to the line equals the circle's radius.

Step-by-Step Solution

Step 1: Define the Circle's Center and Constraints

• We are given that the center $(h, k)$ lies on the line $x + y = 2$. We want to express $k$ in terms of $h$ to reduce the number of variables.
• Since $(h, k)$ lies on $x + y = 2$, we have $h + k = 2$. Thus, $k = 2 - h$. The center can be represented as $(h, 2 - h)$.
• We are given that the center lies in the first quadrant. This means both coordinates must be positive.
• For the center to be in the first quadrant:
  • $h > 0$
  • $2 - h > 0 \implies h < 2$
• Combining these inequalities, we get $0 < h < 2$. This constraint on $h$ is important for verifying the final answer.

Step 2: Apply Tangency Conditions to Find the Radius

• The circle touches the lines $x = 3$ and $y = 2$. The distance from the center $(h, 2-h)$ to each line must equal the radius $r$.

• Condition 1: Circle touches the line $x = 3$

  • The line $x = 3$ can be written as $x - 3 = 0$.
  • Using the distance formula from the point $(h, 2 - h)$ to the line $x - 3 = 0$: $r = \frac{|1 \cdot h + 0 \cdot (2 - h) - 3|}{\sqrt{1^2 + 0^2}}$ $r = \frac{|h - 3|}{1}$ $r = |h - 3| \quad \text{(Equation 1)}$

• Condition 2: Circle touches the line $y = 2$

  • The line $y = 2$ can be written as $y - 2 = 0$.
  • Using the distance formula from the point $(h, 2 - h)$ to the line $y - 2 = 0$: $r = \frac{|0 \cdot h + 1 \cdot (2 - h) - 2|}{\sqrt{0^2 + 1^2}}$ $r = \frac{|2 - h - 2|}{1}$ $r = |-h|$ $r = |h| \quad \text{(Equation 2)}$

Step 3: Solve for h

• Since both Equation 1 and Equation 2 equal the radius $r$, we can equate them to solve for $h$.
• Equating Equation 1 and Equation 2: $|h - 3| = |h|$
• Squaring both sides to eliminate the absolute values: $(h - 3)^2 = h^2$
• Expanding the left side: $h^2 - 6h + 9 = h^2$
• Subtracting $h^2$ from both sides: $-6h + 9 = 0$
• Solving for $h$: $6h = 9$ $h = \frac{9}{6} = \frac{3}{2}$

Step 4: Verify the Quadrant Condition and Calculate the Radius

• We need to check if $h = \frac{3}{2}$ satisfies the first quadrant condition, $0 < h < 2$.
• Since $0 < \frac{3}{2} < 2$, the value $h = \frac{3}{2}$ is valid.
• Now, we can find the radius $r$ using Equation 2: $r = |h| = \left|\frac{3}{2}\right| = \frac{3}{2}$
• Therefore, the radius of the circle is $r = \frac{3}{2}$.

Step 5: Calculate the Diameter

• The diameter is twice the radius.
• Diameter $= 2r = 2 \times \frac{3}{2} = 3$

Common Mistakes & Tips

• Always consider absolute values: Distance must be non-negative.
• Quadrant conditions are essential: They help eliminate extraneous solutions.
• Parameterization simplifies: Reduces the number of variables and makes the problem easier to solve.

Summary

By expressing the center of the circle in terms of a single variable and using the tangency conditions with the given lines, we were able to set up equations to solve for the radius. We then doubled the radius to find the diameter of the circle. The diameter of the circle is 3.

The final answer is \boxed{3}.