Key Concepts and Formulas

• The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, with center $(-g, -f)$ and radius $r = \sqrt{g^2 + f^2 - c}$.
• The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• The number of common tangents depends on the relationship between the distance between the centers ($C_1C_2$) and the sum and difference of the radii ($r_1$ and $r_2$). If $C_1C_2 = r_1 + r_2$, the circles touch externally, and there are 3 common tangents.

Step-by-Step Solution

Step 1: Determine the Centers and Radii of the Circles

We need to find the center and radius of each circle using the general equation of a circle. This allows us to determine the relative positions of the circles.

For the first circle: The given equation is $x^2 + y^2 - 4x - 6y - 12 = 0$. Comparing with the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$, we have: $2g = -4 \Rightarrow g = -2$ $2f = -6 \Rightarrow f = -3$ $c = -12$

Therefore, the center of the first circle, $C_1$, is $(-g, -f) = (2, 3)$. The radius of the first circle, $r_1$, is $\sqrt{g^2 + f^2 - c} = \sqrt{(-2)^2 + (-3)^2 - (-12)} = \sqrt{4 + 9 + 12} = \sqrt{25} = 5$. $C_1 = (2, 3), \quad r_1 = 5$

For the second circle: The given equation is $x^2 + y^2 + 6x + 18y + 26 = 0$. Comparing with the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$, we have: $2g = 6 \Rightarrow g = 3$ $2f = 18 \Rightarrow f = 9$ $c = 26$

Therefore, the center of the second circle, $C_2$, is $(-g, -f) = (-3, -9)$. The radius of the second circle, $r_2$, is $\sqrt{g^2 + f^2 - c} = \sqrt{(3)^2 + (9)^2 - (26)} = \sqrt{9 + 81 - 26} = \sqrt{64} = 8$. $C_2 = (-3, -9), \quad r_2 = 8$

Step 2: Calculate the Distance Between the Centers

We need to calculate the distance between the centers $C_1(2, 3)$ and $C_2(-3, -9)$ using the distance formula. This distance is crucial for determining the relative positions of the two circles.

The distance $C_1C_2$ is: $C_1C_2 = \sqrt{(-3 - 2)^2 + (-9 - 3)^2} = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

$C_1C_2 = 13$

Step 3: Compare the Distance Between Centers with the Radii

We need to compare the distance between the centers, $C_1C_2$, with the sum and difference of the radii, $r_1 + r_2$ and $|r_1 - r_2|$, to determine the relative positions of the circles.

We have: $C_1C_2 = 13$ $r_1 = 5$ $r_2 = 8$

The sum of the radii is: $r_1 + r_2 = 5 + 8 = 13$

The absolute difference of the radii is: $|r_1 - r_2| = |5 - 8| = |-3| = 3$

We observe that $C_1C_2 = r_1 + r_2 = 13$.

Step 4: Determine the Number of Common Tangents

Since $C_1C_2 = r_1 + r_2$, the two circles touch externally. When two circles touch externally, they have 3 common tangents.

Common Mistakes & Tips

• Always double-check the signs when calculating the center and radius from the general equation of a circle. A sign error can lead to an incorrect distance calculation and a wrong conclusion.
• Remember to take the square root when calculating the radius. Forgetting this step is a common mistake.
• Carefully compare the distance between the centers with both the sum and the absolute difference of the radii to correctly determine the relative positions of the circles.

Summary

We determined the centers and radii of the two circles, calculated the distance between their centers, and compared this distance with the sum and difference of their radii. Since the distance between the centers is equal to the sum of the radii, the circles touch externally, and there are 3 common tangents.

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