<s>[INST] Find all circles given two points and not the center This is probably pretty straight forward for you.
I have two points on a circle, $(-4, 7)$ and $(-5, 0)$. Given these two points and the radius $5$, what are all the possible equations?
My first idea was to solve the system
$\sqrt{(-4-a)^2 + (7-b)^2} = 5\tag 1$
$\sqrt{(-5-a)^2 + b^2} = 5\tag 2$ 
Where $(a, b)$ is the center point. However, I don't get the correct answers. Maybe this question should be about how to solve the system.
Thanks in advance!
Edit:
This is the formulation of the problem:
Find all circles through the points $(-4, 7)$ and $(-5, 0)$ with a radius of $5$.
 [/INST] I'd personally start with the following 2 equations.
$(-4-a)^2 + (7-b)^2 = 25$ (1)
$(-5-a)^2 + b^2 = 25$ (2)
distribute out the squares.
$16 + 8a +a^2 + 49 - 14b + b^2 = 25$ (1)
$a^2 + b^2 + 8a - 14b + 40 = 0$ (1)
$25 + 10a + a^2 + b^2 = 25$ (2)
$a^2 + b^2 + 10a = 0$ now we set them equal
$a^2 + b^2 + 10a = a^2 + b^2 + 8a - 14b + 40$
$2a = -14b + 40$
$a = 20 -7b$ now we plug this into an earlier equation. I'm using equation 2
$(20 - 7b)^2 + b^2 + 10(20 - 7b) = 0$
$400 - 280b + 49b^2 + b^2 + 200 - 70b = 0$
$600 -350b + 50b^2 = 0$
Using the quadratic formula, b equals 4 or 3, so a = -8 or -1 respectively, giving us two possible centers of (-8,4) or (-1,3). This can be verified using the original equation.
</s>