18. Two circles $C_{1}$ and $C_{2}$ have their centres at the point $(3,4)$ and touch a third circle, $C_{3}$. The centre of $C_{3}$ is at the point $(0,0)$ and its radius is 2 .
What is the sum of the radii of the two circles $C_{1}$ and $C_{2}$ ?
A 6
B 7
C 8
D 9
E 10
Solution
E

Let $P$ be the centre of the circles $C_{1}$ and $C_{2}, O$ be the centre of the circle $C_{3}$, and $R$ and $S$ be the points where $C_{1}$ and $C_{2}$, respectively, touch $C_{3}$. We let the radius of $C_{1}$ be $r$.
Note that the points $O, P, R$ and $S$ lie on a straight line.
By Pythagoras' Theorem, the distance between the point $P$ with coordinates $(3,4)$ and the point $O$ with coordinates $(0,0)$ is $\sqrt{3^{2}+4^{2}}=\sqrt{25}=5$.
Therefore, $r+2=5$ and hence $r=3$.
The radius of the circle $C_{2}$ is $r+2+2=3+2+2=7$.
Therefore the sum of the radii of $C_{1}$ and $C_{2}$ is $3+7=10$.
For inVESTIGATION
18.1 Prove that the points $P, R, O$ and $S$ lie on a straight line.