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MHT CET202527 Apr 2025Evening ShiftMathematicsCircleActual

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

Options

  1. A1
  2. B2
  3. C3
  4. D4

Correct answer

A. 1

Step-by-step solution

The centers and radii of the circles are determined by completing the square or comparing with the general circle equation. For S₁: x^2+y^2-4 x-6 y-12=0 , center is C₁ = (2, 3) and radius r₁ = 5 . For S₂: x^2+y^2+6 x+18 y+26=0 , center is C₂ = (-3, -9) and radius r₂ = 8 . The distance between centers is d = (-3-2)^2 + (-9-3)^2 = 13 . Since d = r₁ + r₂ = 5 + 8 = 13 , the circles touch externally. Two externally tangent circles have exactly three common tangents. Answer: 3

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