Key Concepts and Formulas

• The equation of a circle with center $(h, k)$ and radius $r$ is $(x-h)^2 + (y-k)^2 = r^2$.
• The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Two circles with radii $R_1$ and $R_2$ and distance $d$ between their centers intersect at exactly two points if $|R_1 - R_2| < d < R_1 + R_2$.

Step-by-Step Solution

Step 1: Determine the properties of the first circle, C.

We are given that circle C has a radius of 2, lies in the second quadrant, and touches both coordinate axes. This means its center must be at $(-2, 2)$ because the coordinates of the center are $(-R, R)$ in the second quadrant, where $R$ is the radius. Therefore, the center of the circle $C_1$ is $(-2, 2)$ and its radius $R_1 = 2$. The equation of circle C is $(x - (-2))^2 + (y - 2)^2 = 2^2$, which simplifies to $(x+2)^2 + (y-2)^2 = 4$.

Step 2: Determine the properties of the second circle.

The second circle has its center at $(2, 5)$ and its radius is $r$. Thus, the center of the second circle $C_2$ is $(2, 5)$ and its radius $R_2 = r$. The equation of the second circle is $(x-2)^2 + (y-5)^2 = r^2$.

Step 3: Calculate the distance between the centers of the two circles.

Using the distance formula, the distance $d$ between $C_1(-2, 2)$ and $C_2(2, 5)$ is: $d = \sqrt{(2 - (-2))^2 + (5 - 2)^2} = \sqrt{(2+2)^2 + (5-2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$ So, the distance between the centers is $d=5$.

Step 4: Apply the condition for intersection at exactly two points.

For the two circles to intersect at exactly two points, the distance between their centers must satisfy the condition: $|R_1 - R_2| < d < R_1 + R_2$. Substituting $R_1 = 2$, $R_2 = r$, and $d = 5$, we have: $|2 - r| < 5 < 2 + r$ This is a compound inequality that can be separated into two inequalities:

1. $|2 - r| < 5$
2. $5 < 2 + r$

Step 5: Solve the inequalities for r.

• Solving the first inequality: $|2 - r| < 5$ $-5 < 2 - r < 5$ Subtract 2 from all parts: $-7 < -r < 3$ Multiply by -1 and reverse the inequality signs: $7 > r > -3$ $-3 < r < 7$

• Solving the second inequality: $5 < 2 + r$ Subtract 2 from both sides: $3 < r$

Step 6: Combine the solutions to find the possible values of r.

We need to find the values of $r$ that satisfy both $-3 < r < 7$ and $3 < r$. The intersection of the intervals $(-3, 7)$ and $(3, \infty)$ is $(3, 7)$. Therefore, $3 < r < 7$.

Step 7: Determine $\alpha$ and $\beta$ and calculate the final expression.

The possible values of $r$ are in the interval $(\alpha, \beta)$, so $\alpha = 3$ and $\beta = 7$. Now, we calculate $3\beta - 2\alpha$: $3\beta - 2\alpha = 3(7) - 2(3) = 21 - 6 = 15$

Common Mistakes & Tips

• Be careful with signs when determining the center of the circle in the second quadrant. The center is at $(-R, R)$ where $R$ is the radius.
• Remember to reverse the inequality signs when multiplying or dividing by a negative number.
• Ensure you correctly find the intersection of the intervals when combining inequalities.

Summary

By correctly identifying the circle properties, calculating the distance between the centers, and carefully solving the inequalities, we found the range of possible radii to be $3 < r < 7$. Therefore, $\alpha = 3$ and $\beta = 7$, and $3\beta - 2\alpha = 15$.

The final answer is \boxed{15}, which corresponds to option (D).