Relative position of two Circles
Let S, S' be two (nonconcentric) circles with centers A, B and radii r_{1}
, r_{2} and d be the distance between their centers, then
(i) One circle lies completely inside the other iff d <  r_{1} r_{2}

(ii) The two circles touch internally iff d =  r_{1} r_{2} 
(iii) The two circles intersect in two points iff d > r_{1} r_{2}
and d < r_{1} +r_{2
}(iv) The two circles touch externally iff d = r_{1} +r_{2
}(v) One circle lies completely outside the other iff d > r_{1}
+r_{2}
Note
1. If two circles intersect, then we can solve the equations of the circles simultaneously to find the points of intersection. In particular, the equation of the common chord is given by S S' = 0.2. If the two circles touch (internally or externally), then the equation of their common tangent is given by S S' = 0.
Illustrative Examples
Example
Show that the circles x² +y² 2 x = 0 and x² +y² +6 x
6 y +2 = 0 touch each other. Do these circles touch externally or internally?
Find the point of contact and the common tangent.
Solution
The equations of the given circles are
S = x² + y² 2 x = 0
...(i)
and S = x² + y²+6x 6y +2 = 0 ...(ii)
Their centers are A (1, 0) and B (3, 3), and their radii are
r_{1} = [1² +0² 0] = 1 and r_{2}
= [9 +9 2] = 4 respectively
The distance between their centers
= d = [( 3 1)² +(3 0)²] = 5 = 1 +4
=> d = r_{1} + r_{2
}=> the given circles (i) and (ii) touch externally and the point of
contact P divides [AB] internally in the ratio r_{1} : r_{2}
i.e. in the ratio 1 : 4
The coordinates of the point of contact are
(1.(3) +4.1)/(1+4), (1.3 +4.0)/(1+4) i.e. (1/5, 3/5)
The equation of the common tangent is S S' = 0
=> 8 x +6 y 2 = 0 => 4 x 3 y +1 = 0
Exercise
 Prove that the circle x² + y² 6 x 2 y +9 = 0 lies entirely inside the circle x² + y² = 18.
 Prove that the circles x² + y² 4 x +6 y +8 = 0 and x² + y² 10 x 6 y +14 = 0 touch each other externally. Find their point of contact and also the common tangent.
 Show that the circles x² + y² +2 x 6 y +9 = 0 and x² + y² +8 x 6 y +9 = 0 touch internally. Find their point of contact and also the common tangent.
 Prove that the circles x² + y² 6x 2 y +1 = 0 and x² + y² +2 x 8 y +13 = 0 touch one another and find the equation of the tangent at their point of contact.
 Show that the circles x² + y² = 2 and x² + y² 6x 6 y +10 = 0 touch each other. Do these circles touch externally or internally? Also find their point of contact.
 Find the equation of the circle whose radius is 3 and which touches the circle x² + y² 4 x 6 y 12 = 0 internally at the point (1, 1).
Answers
2. (3, 1, x +2 y 1 = 03. (0, 3), x = 0
4. 4 x 3 y +6 = 0
5. Externally; (1, 1)
6. 5 (x² + y²) 8x 14 y +32 = 0