Number of Common Tangents to Two Circles


This lesson will talk about number and equations of common tangents to two given circles. I’ll divide the lessons into two parts – this one talking about the number, and the next one about equations. This will be followed by a few examples in the subsequent lesson.

The number of common tangents will depend upon the relative position of the two circles – something which I talked about in the previous lesson. It is pretty much intuitive, by having a look at the circles. We can also prove it using algebra by using equations of tangents – which I’ll try to cover in the next part of the lesson. Let’s go back to all the cases then.

1. Two circles lying outside each other

In this case, there will be 4 common tangents, as shown in the figure.

circle common tangents

The tangents intersecting between the circles are known as transverse common tangents, and the other two are referred to as the direct common tangents.

2. Two circles touching each other externally

In this case, there will be 3 common tangents, as shown in the figure.

circle common tangents

The tangent in between can be thought of as the transverse tangents in the previous case coinciding together

3. Two circles intersecting each other at two points

In this case, there will be 2 common tangents, as shown in the figure.

circle common tangents

4. Two circles touching each other internally

In this case, there will be only 1 common tangent, as shown in the figure.

Think of is as the two tangents from the previous case coinciding into one, when one of the circles is pushed towards the other.

5. One circle lying inside another

In this case, there will not be any common tangent, as any line touching the inner circle will always cut the outer circle at two points. Have a look at the figure.

circle common tangents

I hope you’re convinced about the number of common tangents. In the next part, I’ll talk about their equations.

Lesson Summary

I’ve summarised the lesson in the following table:

Position

Number of Common Tangents

Lying outside each other

4

Touching externally

3

Intersecting at two points

2

Touching internally

1

One lying inside other

0