Question

Two circles are given in a plane. State the conditions under which there is a common tangent line to both.
Explain how to construct the common tangent. Prove that your construction is correct.

Answer 1

Conditions under which there is a common tangent line to both the circles are:

1. When two circles touch each other internally 1 common tangent can be drawn to the circles i.e.  |r₁ - r₂| = c₁c₂

2. When two circles intersect in two real and distinct points, 2 common tangents can be drawn to the circles i.e. |r₁ - r₂| < c₁c₂ < r₁ + r₂

3. When two circles touch each other externally, 3 common tangents can be drawn to the circles i.e.  r₁ + r₂ = c₁c₂

4. When two circle neither touch nor intersect and one lies outside the other, then 4 common tangents can be drawn i.e.  r₁ + r₂ < c₁c₂