Expand $(x - h)^2 + (y - k)^2 = r^2$ and you get an equation like $x^2 + y^2 + Dx + Ey + F = 0$. To read the center and radius from that expanded form, run the process backward with completing the square from Completing the Square, then use Equation of a Circle.
Por qué funciona
For any number $D$, $\left(x + \frac{D}{2}\right)^2 = x^2 + Dx + \frac{D^2}{4}$. So $x^2 + Dx$ is a perfect square except for the missing $\frac{D^2}{4}$. Adding $\left(\frac{D}{2}\right)^2$ completes it. Adding it to one side would break the equation, so add the same amount to the other side too.
Idea clave
Group the $x$ terms and the $y$ terms, move the constant to the right, add $\left(\frac{D}{2}\right)^2$ and $\left(\frac{E}{2}\right)^2$ to both sides, and factor: $(x - h)^2 + (y - k)^2 = r^2$.
Ejemplo resuelto: Both variables
Find the center and radius of $x^2 + y^2 - 6x + 4y - 12 = 0$.
- Group and move the constant: $(x^2 - 6x) + (y^2 + 4y) = 12$.
- Half of $-6$ is $-3$, squared is $9$; half of $4$ is $2$, squared is $4$. Add both to both sides: $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$.
- Factor: $(x - 3)^2 + (y + 2)^2 = 25$.
Respuesta: Center $(3, -2)$, radius $5$
Ejemplo resuelto: A leading coefficient
Find the center and radius of $2x^2 + 2y^2 + 8x - 20y + 40 = 0$.
- Divide everything by 2 first: $x^2 + y^2 + 4x - 10y + 20 = 0$.
- $(x^2 + 4x + 4) + (y^2 - 10y + 25)$ $= -20 + 4 + 25$, so the right side is $9$.
- $(x + 2)^2 + (y - 5)^2 = 9$.
Respuesta: Center $(-2, 5)$, radius $3$
Ejemplo resuelto: Not a circle
What is the graph of $x^2 + y^2 - 2x + 6y + 15 = 0$?
- $(x^2 - 2x + 1) + (y^2 + 6y + 9)$ $= -15 + 1 + 9$, so the right side is $-5$.
- $(x - 1)^2 + (y + 3)^2 = -5$. A sum of squares cannot be negative.
Respuesta: No points at all: it is not a circle.
Error común
Adding the completing numbers to one side only. In the first example, writing $(x - 3)^2 + (y + 2)^2 = 12$ forgets that $9$ and $4$ were added on the left. Check by expanding your answer: it should give back the original equation.