Here is a pot, balanced on a pair of steel slabs that meet at right angles below the pot. The intersection of the slabs is treated here as the origin of a set of co-ordinate axes which hit the edge of the pot at the distances $-6, 15,$ and $8$ as shown.

What is the diameter of the pot?

## Solution

The problem gives, in effect. three points on a circle as in the figure below. As three points determine a circle, it should be possible to find the equation of this one:

$(x-h)^2 + (y-k)^2 = r^2,$

where $(h, k)$ is the location of the center, and so determine the radius $r$. Plugging in the three points gives the three equations:

$\begin{aligned} 36 + 12 h + h^2 + k^2 &=& r^2, \\ 225 - 30 h + h^2 + k^2 &=& r^2, \\ h^2 + 64 - 16 k + k^2 &=& r^2. \end{aligned}$Subtracting the equations in pairs gives

$\begin{aligned} 42 h - 189 &=& 0, \\ 36 + 12 h - 64 - 16k &=& 0. \end{aligned}$These two linear equations are easily solved. It turns out that $h = 189/42 = 9/2$ and $k = -26/16 = -13/8$. Plugging this information into any of the first three equations, we discover that $r = 85/4 = 21.25.$