# Different Cases of Two Circles

Different cases of intersection of two circles:

Let the two circles be

(

*x*âˆ’*x*_{1})^{2}+ (*y*âˆ’*y*_{1})^{2}=*r*_{1}^{2 }...(9)and (

*x*âˆ’*x*_{2})^{2}+ (*y*âˆ’*y*_{2})^{2}=*r*_{2}^{2 }...(10)with centers

*C*_{1}(*x*_{1},*y*_{1}) and*C*_{2}(*x*_{2},*y*_{2}) and radii*r*_{1}and*r*_{2}respectively. Then the following cases may arise:**Condition:**|

*C*

_{1}

*C*

_{2}| >

*r*

_{1}+

*r*

_{2}

Number of common tangents: Four Coordinates of

*T*areCoordinates of

*D*are**Condition :**|

*C*

_{1}

*C*

_{2}| =

*r*

_{1}+

*r*

_{2}

Number of common tangents : Three Coordinates of

*T*areCoordinates of

*D*areRadical axis : Common tangent at

*T***Condition:**|

*r*

_{1}â€“

*r*

_{2}| <|

*C*

_{1}

*C*

_{2}| <

*r*

_{1}+

*r*

_{2}

Number of common tangents: Two Coordinates of

*D*areRadical axis: Common chord

**Condition:**

*C*

_{1}

*C*

_{2}= |

*r*

_{1}â€“

*r*

_{2}|

Number of common tangents: One Coordinates of

*P*areRadical axis: Common tangent at

*P***Condition:**|

*C*

_{1}

*C*

_{2}| <

*r*

_{1}âˆ’

*r*

_{2}|

No common tangent

# Angle of intersection of two circles

Angle between two circles is given by

cos

*Î¸**=*where

*r*_{1},*r*_{2 }are radii of circles and*d*is distance between their centers. If circles intersecting orthogonally then*r*_{1}^{2}+*r*_{2}^{2}=*d*^{2}. If circle equations are*x*^{2}+*y*^{2}+ 2*gx*+ 2*fy*+*c*= 0 and*x*^{2}+*y*^{2}+ 2*g*_{1}*x*+ 2*f*_{1}*y*+*c*_{1}= 0, then we have 2*gg*_{1}+ 2*ff*_{1}=*c*+*c*_{1}.# Family of circles

- The equation of the family of circles passing through the point of intersection of two given circles
*S*= 0 and*S*â€² = 0 is given as*S*+*Î»**S*â€² = 0, where*Î»*is a parameter,*Î»*Ï€ â€“1. - 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*+*Î»**L*= 0, where*Î»*is a parameter. - 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*+*Î»**L*= 0, where*Î»*is a parameter. - The equation of a family of circles passing through two given points
*P*(*x*_{1},*y*_{1}) and*Q*(*x*_{2},*y*_{2}) can be written in the form*x*â€“*x*_{1}) (*x*â€“*x*_{2}) + (*y*â€“*y*_{1})(*y*â€“*y*_{2}) +*Î»*= 0*Î»*is a parameter. - The equation of family of circles which touch
*y*â€“*y*_{1}=*m*(*x*â€“*x*_{1}) at (*x*_{1},*y*_{1}) for any finite*m*is (*x*â€“*x*_{1})^{2}+ (*y*â€“*y*_{1})^{2}+*Î»*{(*y*â€“*y*_{1}) â€“*m*(*x*â€“*x*_{1})} = 0 and if*m*is infinite, the family of circles is (*x*â€“*x*_{1})^{2}+ (*y*â€“*y*_{1})^{2}+*Î»*(*x*â€“*x*_{1}) = 0, where*Î»*is a parameter and, (*x*â€“*x*_{1})^{2}+ (*y*â€“*y*_{1})^{2}= 0 is point circle at point (*x*_{1},*y*_{1}).