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Different Cases of Two Circles

Different cases of intersection of two circles:
 
Let the two circles be
 
(x x1)2 + (y y1)2 = r12  ...(9)
and (x x2)2 + (y y2)2 = r22  ...(10)
 
with centers C1(x1, y1) and C2(x2, y2) and radii r1 and r2 respectively. Then the following cases may arise:
 
Condition: |C1C2| > r1 + r2
 
Number of common tangents: Four Coordinates of T are
72647.png
 
Coordinates of D are
72655.png
 
72665.png
 
Condition : |C1C2 | = r1 + r2
 
Number of common tangents : Three Coordinates of T are
72671.png
 
Coordinates of D are
72682.png
 
Radical axis : Common tangent at T
 
72689.png
 
Condition: |r1r2| <|C1C2 | < r1 + r2
 
Number of common tangents: Two Coordinates of D are
72696.png
 
Radical axis: Common chord
 
72702.png
 
Condition: C1C2 = |r1r2|
 
Number of common tangents: One Coordinates of P are
72709.png
 
Radical axis: Common tangent at P
 
72720.png
 
Condition: |C1C2 | < r1 r2|
 
No common tangent
 
72734.png

Angle of intersection of two circles

Angle between two circles is given by
cos θ = 72742.png
 
where r1, r2 are radii of circles and d is distance between their centers. If circles intersecting orthogonally then r12 + r22 = d2. If circle equations are x2 + y2 + 2gx + 2fy + c = 0 and x2 + y2 + 2g1x + 2f1y + c1 = 0, then we have 2gg1 + 2ff1 = c + c1.

Family of circles

  1. The equation of the family of circles passing through the point of intersection of two given circles S = 0 and S = 0 is given as S + λS = 0, where λ is a parameter, λ π –1.
     
    68226.png
  2. The equation of the family of circles passing through the point of intersection of circle S = 0 and a line L = 0 is given as S + λL = 0, where λ is a parameter.
     
    68220.png
  3. The equation of the family of circles touching the circle S = 0 and the line L = 0 at their point of contact P is S + λL = 0, where λ is a parameter.
     
    68214.png
  4. The equation of a family of circles passing through two given points P (x1, y1) and Q (x2, y2) can be written in the form
     
    (xx1) (xx2) + (yy1)(yy2) + λ 68208.png = 0
     
    where λ is a parameter.
     
    68202.png
  5. The equation of family of circles which touch yy1 = m (xx1) at (x1, y1) for any finite m is (xx1)2 + (yy1)2 + λ {(yy1) – m (xx1)} = 0 and if m is infinite, the family of circles is (xx1)2 + (yy1)2 + λ (xx1) = 0, where λ is a parameter and, (xx1)2 + (yy1)2 = 0 is point circle at point (x1, y1).




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