1. Suppose that $\pi$ is a plane and $A$ and $B$ are two points on the plane $\pi$. If the distance between $A$ and $B$ is $33 \mathrm{~cm}$, how many lines are there in the plane such that the distance between each line and $A$ is $7 \mathrm{~cm}$ and the distance between each line and $B$ is $26 \mathrm{~cm}$ respectively?
(A) 1
(B) 2
(C) 3
(D) 4
(E) Infinitely many
1. Answer: (C)
In the plane $\pi$, draw a circle of radius $7 \mathrm{~cm}$ centred at $A$ and a circle of radius $26 \mathrm{~cm}$ centred at $B$.

If $\ell$ is a line on the plane $\pi$, and the distance between $\ell$ and $A$ is $7 \mathrm{~cm}$ and the distance between $\ell$ and $B$ is $26 \mathrm{~cm}$, then $\ell$ must be tangential to both circles. Clearly, there are 3 lines in the plane that are tangential to both circles, as shown in the figure above.