Let the two circles be
(x-h1)² + (y-k1)² = a²
(x-h2)² + (y-k2)² = a²
with centres C1(h1,k1) and C2(h2,k2) and radii a1 and a2 respectively.
The various cases that can occur are
Case 1. When C1C2>a1+a2 i.e., the distance between the centres is greater than the sum of radii.
In this case, the circles do not intersect each other and four common tangents can be drawn to two circles.
Two of them are direct common tangents. Two are transverse common tangents.
The intersection between common tangents (T2) lies on the line joining C1 and C2 and divides it externally in the ratio a1/a2. C1T2/CTs = a1/a2
The intersection between transverse tangents (T1) lies on the line joining C1 and C2 and divides the line internally in the ration a1/a2. i.e., C1T1/C2T1 = a1/a2.
Case 2. When C1C2 = a1+a2 i.e, the distance between the centres of circles is equal to the sum of the radii, two direct tangents are real and distinct, but the transverse tangents are coincident.
Case 3. When C1C2
Case 4. When C1C2 = a1-a2 i.e., the distance between the centres is equal to the difference of the radii.
In this case two tangents are real and coincident while the other two tangents are imaginary.
Case 5. When C1C2 < a1-a2 i.e., the distance between the centres is less than the difference of the radii.
In this case all four common tangents are imaginary.