Multiple choice

The number of common tangents to the circles x2 + y2 – 4x – 6y – 12 = 0 and x2 + y2 + 6x + 18y + 26 = 0, is

  1. 1

  2. 2

  3. 3

  4. 4
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Circle 1: center (2, 3), radius = sqrt(4+9+12) = 5. Circle 2: center (-3, -9), radius = sqrt(9+81-26) = 8. Distance between centers = sqrt((2 - -3)^2 + (3 - -9)^2) = sqrt(5^2 + 12^2) = 13. Since distance (13) = r1 + r2 (5 + 8), the circles touch externally. They have 3 common tangents.

AI explanation

The centers of the two circles are found by completing the square, giving C1 at (2, 3) with a radius of 5 cm, and C2 at (-3, -9) with a radius of 8 cm. The distance between centers is calculated using the distance formula as the square root of 25 plus 144, which equals 13 cm. Because the distance between the centers equals the sum of their radii, the two circles touch externally, meaning the number of common tangents is 3.

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