what is the circle with the smallest radius that contains the three points $(0,0)$ , $(1,1)$, and $(2,3)$?

Question

How to write a model that gives us a circle with the smallest radius that contain three points $(0,0)$ , $(1,1)$, and $(2,3)$?

I've tried to model this as following:

if $x, y$ be the location and $r$ is radius:

$ Min \ \ x^2+y^2 = r^2$

$ x^2+y^2 \le 0$

$ (x-1)^2+(y-1)^2 \le 1$

$ (x-2)^2+(y-3)^2 \le \sqrt 13$

Answer 1

How about:

min $r^2$ subject to:

$(x-0)^2 + (y-0)^2 \leq r^2$

$(x-1)^2 + (y-1)^2 \leq r^2$

$(x-2)^2 + (y-3)^2 \leq r^2$

Answer 2

• Consider the points pair-wise, call them $A$ and $B$. Let the other point be $C$

• Consider the circle with $AB$ as diameter and check if $C$ is within this circle.

If you consider all three possibilities, you'll find that the line joining $(0, 0)$ and $(2, 3)$ is the diameter of the required circle. The equation of the circle is

$$(x - 1)^2 + (y - \frac{3}{2})^2 = \left(\frac{\sqrt{13}}{2} \right)^2$$

It is easy to verify that the other point $(1,1)$ lies inside the above circle.