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.
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.
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.
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.
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 |