Appolonius three circles problem, finding centre of tangent circle (analytic geometry)

Question

Given 3 circles:

• $C_1$ centered at $(0,0)$ with radius 1
• $C_2$ centered at $(a,0)$ with radius $a+1$
• $C_3$ centered at $(-a,0)$ with radius $a+1$

(so $C_1$ is internaly tangent to both $C_2$ and $C_3$ )

Question:

What are the centres of circle $C_4$ and $C_5$ that are tangent to all three circles?

I know they are on the $x=0$ axis but what is their $y$ coordinate?

Answer 1

WOLOG, we will assume $C_4$ / $C_5$ are lying in upper / lower half-plane respectively.

Let $c_i$ and $r_i$ be the center and radius for circle $C_i$ where $i = 1,\ldots,5$.
By symmetry, $r_4 = r_5$. Let us call this common value $r$. In term of $r$, $$c_4 = (0,1+r)\quad\text{ and }\quad c_5 = (0,-1-r)$$

Since $C_4$ is inside and tangent to $C_2$, we have

$$|c_4 - c_2| = r_2 - r_4 \implies (r+1)^2 + a^2 = (a+1-r)^2 \implies r = \frac{a}{a+2} $$ This leads to $c_4 = -c_5 = \left( 0, 2\left(\frac{a+1}{a+2}\right)\right)$.