Find the equations of all three circles

Question

Circles A, B, and C are positioned in Quadrant I as follows:

Circle $A$ contains the points $(1,0)$ and $(0,1)$. Circle $B$ rests on the x-axis to the right of Circle $A$ and is tangent to Circle $A$. Circle $C$ also rests on the x-axis, occupies the small space between Circles $A$ and $B$, and is tangent to both. If the radius of Circle $B$ is $2$, find the equations of all three circles.

I know that circle $A$ is $(x-1)^2+(y-1)^2=1$. I also know that circle $B$ is in the form $(x-k)^2+(y-2)^2=4$. Now I'm stuck...

Answer 1

The center of your first circle is $(1,1)$ and its radius is $1$, of course. The position of the center of the right-most circle is $(y_0,2)$ and its radius $2$, for some unknown $y_0$. This single variable is trivial to compute from the Pythagorean theorem:

$$(y_0 - 1)^2 + (2 - 1)^2 = 3$$

The center of the final circle is $(x_1, y_1)$ and radius $y_1$, also from the Pythagorean theorem.

The lines segments connecting the centers have lengths equal to the sum of the radii. This gives you a set of equations that can be solved directly for the positions of the centers and the unknown radii.

Answer 2

Don't know if it's worth posting. To construct the circles:

1. The first circle's center lays on the perpendicular bisector to $AB$ (sorry for mixing labels) with $A=(0,1),\,B=(1,0)$. If it's not given that the first circle touches the $x$-axis, then the center of the first circle $O$ can be anywhere on the perpendicular bisector to $AB$.
   
2. If we translate the $x$-axis down by the radius of the first circle and build the parabola with the focus $O$ and the directrix of this translated line, then the center of the second circle will lie somewhere on that parabola. We also know the radius of $2$ so we can derive the circle equation.
3. Similarly if we translate the $x$-axis down by $2$ (the radius of the second circle) units and use the line as the directrix of the right parabola with focus in the center of the second circle ($D$) then
4. the parabolas will intersect in the center of the third circle. Viola. You just need to write all the equations down in symbols.

Answer 3

Considering that the three circles are all three tangents to the abscis axis and that they are tangent to each other, this gives us the conditions that allow us to write the equations of the circles:

A) $x^{2}+y^{2}-2x-2y+1=0$,

B) $x^{2}+y^{2}-2(2\sqrt{2}+1)x-4y+4\sqrt{2}+9=0$,

C) $x^{2}+y^{2}-2(5-2\sqrt{2})x-4(3-2\sqrt{2})y+33-20\sqrt{2}=0$.