Equation of circle touching 2 circles and x-axis

Question

You are given 2 circles with center 
(2,2) and (8,4) respectively. 
Both the circles touch x-axis. 

You need to find the equation of a circle which touches these 2 circles and the x-axis as well. I tried using the equal distance property, and also equating the tangent equations of the 3rd circle with each one of them, but couldn't get any solution. Any help would be appreciated.

Answer 1

Let $C_1,C_2$ be the circles centered at $P_1=(2,2),P_2=(8,4)$, respectively, with each circle tangent to the $x$-axis.

Let $C_3$ be the circle tangent to both $C_1$ and $C_2$, and also tangent to the $x$-axis.

Since each of the three circles is tangent to the $x$-axis, it follows that for each circle, the radius is equal to the $y$-coordinate of the circle's center.

Thus,

• $C_1$ has radius $2$.$\\[4pt]$
• $C_2$ has radius $4$.$\\[4pt]$
• $C_3$ has center $(h,r)$, where $r$ is the radius, and $h,r$ are yet to be determined.

Since $C_3$ is tangent to $C_1$, it follows that $d(P_3,P_1)=r+2$.

Since $C_3$ is tangent to $C_2$, it follows that $d(P_3,P_2)=r+4$.

Hence, by the distance formula, we get \begin{align*} (h-2)^2+(r-2)^2&=(r+2)^2\\[4pt] (h-8)^2+(r-4)^2&=(r+4)^2\\[4pt] \end{align*} Two equations, two unknowns.

Can you finish it?