CAT Common Tangents to Two Circles - Practice Questions & MCQ

Common Tangent: If the same line is tangent to two circles drawn on the same plane, then the line is called a common tangent to the circles. The distance between the point of contact is called the length of the common tangent.

Direct common tangent: If both the circles lie on the same side of the tangent.

Transverse common tangent: If both the circles lie on either side of the tangent.

When two circles are drawn on the same plane with radii r₁ and r₂, with their centers d units apart, then we have the following possibilities.

1.  The two circles are such that one lies inside the other, then |r₁− r₂| > d. In this case, the number of common tangents to the circles is zero.

The two circles are concentric, then d = 0. The points C₁ and C₂ coincide.

2. The two circles may touch each other internally, then |r₁− r₂| = d. In this case, the number of common tangents to the circles is one.

3. The two circles intersect at two points, then |r₁− r₂| < d < r₁ + r₂. In this case, the number of common tangents to the circles is two, i.e., two direct common tangents.

4. The two circles may touch each other externally, then d = r₁ + r₂. In this case, the number of common tangents to the circles is three, i.e., two direct common tangents and one transverse common tangent.

5. The two circles do not meet each other, then d > r₁ + r₂. In this case, the number of common tangents to the circles is four, i.e., 2 direct common tangents and 2 transverse common tangents.

Here is the summary