Consider two circles 𝑀 and 𝑁 with a common tangent 𝐿 at point 𝐴. How many elements are there in the set 𝑀 intersection 𝑁?
Let’s begin by considering two circles and the different ways in which they can intersect. Firstly, if two circles do not meet at all, they have zero points of intersection. Alternatively, they could meet twice and have two points of intersection. Neither of these scenarios satisfies this question though, as we need a common tangent. This is only possible if the two circles touch at one single point, in this question the point 𝐴.
The tangent 𝐿 is a tangent to both of the circles at point 𝐴. This means that these two circles have one point of intersection. And we can therefore conclude that the set 𝑀 intersection 𝑁 has one element.