Angle of Intersection of Two Circles

Angle of Intersection of Two Circles:

1. Equation of Circles:

  • Circles are defined by equations (xh1)2+(yk1)2=r12 and (xh2)2+(yk2)2=r22 with centers (h1,k1) and (h2,k2), and radii r1 and r2 respectively.

2. Geometric Relationship:

  • The angle of intersection between two circles is formed by the tangents drawn from the point of intersection of the circles to their centers.

3. Properties of Tangents:

  • At the point where the circles intersect, the tangents drawn from this point to each circle are perpendicular to the line joining the centers of the circles.

4. Calculation of Angle:

  • Step 1: Determine the Slope of the Line Joining Centers:
    • Calculate the slope m of the line joining the centers using m=k2k1h2h1.
  • Step 2: Calculate the Angle:
    • Use trigonometry to find the angle between the tangents: θ=tan1(m).

5. Example:

  • Circles: C1:(x2)2+(y3)2=4 and C2:(x+1)2+(y1)2=9.
  • Centers: C1 has center (2,3) and C2 has center (1,1).


  • Step 1: Calculate the slope of the line joining centers: m=1312=23.
  • Step 2: Determine the angle formed by the tangents: θ=tan1(23)33.69.