Contact of Two Circles

Contact of Two Circles:

1. Equation of Circles:

  • The equation of a circle with center (h1,k1) and radius r1 is (xh1)2+(yk1)2=r12.
  • Similarly, a circle with center (h2,k2) and radius r2 is (xh2)2+(yk2)2=r22.

2. Geometric Relationship:

  • No Intersection: If the distance between the centers of the circles is greater than the sum of their radii (d>r1+r2), the circles don't intersect.
  • Tangent Circles: When the distance between the centers is equal to the sum of their radii (d=r1+r2), the circles touch externally.
  • Intersecting Circles: If the distance between the centers is less than the sum of their radii (d<r1+r2), the circles intersect at two distinct points.
  • Contained Circle: If one circle lies entirely within the other, the circles are considered to be in contact.

3. Contact Points:

  • Solving for Intersection: Equate the distances between the centers and the radii to determine the points of intersection.
  • d=(h2h1)2+(k2k1)2
  • Case 1: d>r1+r2 → No intersection.
  • Case 2: d=r1+r2 → Tangent circles.
  • Case 3: d<r1+r2 → Intersecting circles.

4. Configuration Analysis:

  • Exterior Contact: Circles externally touch at a single point.
  • Interior Contact: One circle lies completely inside the other, sharing the same center.
  • Partial Overlap: Circles partially intersect, sharing a portion of their circumference.