Question

The number of common tangents to the circles \( x^2 + y^2 - 4x - 6y - 12 = 0 \) and \( x^2 + y^2 + 6x + 18y + 26 = 0 \) is:

• A. \( 2 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( 5 \)

Hint:

The number of common tangents depends on the relative distance between the centers and their radii. For externally touching circles, \( d = r_1 + r_2 \), resulting in 3 common tangents.

Answer 1

The Correct Option is B

Step 1: Find the centers and radii of the two circles.
Circle 1: Center \( C_1 = (2, 3) \), radius \( r_1 = 5 \).
Circle 2: Center \( C_2 = (-3, -9) \), radius \( r_2 = 8 \).

Step 2: Calculate the distance between the centers (\( d \)).
\[ d = \sqrt{(-3 - 2)^2 + (-9 - 3)^2} = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]
Step 3: Compare \( d \) with the sum and difference of the radii.
Sum of radii: \( r_1 + r_2 = 5 + 8 = 13 \)
Difference of radii: \( |r_1 - r_2| = |5 - 8| = 3 \)

Step 4: Determine the number of common tangents.
Since \( d = r_1 + r_2 \) (\( 13 = 13 \)), the two circles touch each other externally. Therefore, there are 3 common tangents.