Question 10
Circles $C_1,C_2$ are tangent to the real number line and externally tangent to each other. Circle $C_1$ touches the line at $1/3$ and has diameter $1/9$; $C_2$ touches it at $3/8$. A third circle $C_3$ is tangent to $C_1,C_2$ and the real line at $x\lt 3/8$. Find $x$ and the radius of $C_3$.
Investigation (4 bonus marks, separate from the 5 base marks): starting with these $C_1,C_2$, define $C_n$ for $n\ge3$ to be tangent to $C_1,C_{n-1}$ and the real line at $x_n\lt x_{n-1}$. For $n\ge4$, find formulas for $x_n$ and its radius $r_n$.
Investigation (4 bonus marks, separate from the 5 base marks): starting with these $C_1,C_2$, define $C_n$ for $n\ge3$ to be tangent to $C_1,C_{n-1}$ and the real line at $x_n\lt x_{n-1}$. For $n\ge4$, find formulas for $x_n$ and its radius $r_n$.