Multiple choice

The number of common tangents to the circles x2 + y2 + 2x + 8y − 25 = 0 and x2 + y2 − 4x − 10y + 19 = 0 is

  1. 1

  2. 2

  3. 3

  4. 4
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B Correct answer
AI explanation

The centers are C1(-1, -4) and C2(2, 5), and their radii are found by completing the square: r1 = sqrt(1+16+25) = sqrt(42) and r2 = sqrt(4+25-19) = sqrt(10). The distance between centers is sqrt((-1-2)^2 + (-4-5)^2) = sqrt(9 + 81) = sqrt(90) = 3 * sqrt(10). Since r1 + r2 is about 10.25 and the distance is about 9.49, the distance between centers is strictly between the difference and the sum of the radii, meaning the circles intersect in two places. Two intersecting circles have exactly 2 common tangents.

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