Circles, ellipses, parabolas and hyperbolas all have equations of the form $Ax^2 + Cy^2 + Dx + Ey + F = 0$ once you multiply everything out. Given such an equation, you can name the conic at a glance, then complete the square to find its center or vertex and match a standard form from parabolas, ellipses or hyperbolas.
Idea clave
For $Ax^2 + Cy^2 + Dx + Ey + F = 0$ (a genuine conic, not a degenerate one): $A = C$: circle; $A$ and $C$ the same sign: ellipse; opposite signs: hyperbola; only one squared term: parabola. If there is an $xy$ term, use $B^2 - 4AC$: negative for an ellipse, zero for a parabola, positive for a hyperbola.
Por qué funciona
Completing the square only shifts the curve; it never changes the coefficients of $x^2$ and $y^2$. So those coefficients already tell you which standard form you will reach: two squares added (ellipse or circle), two squares subtracted (hyperbola), or one square (parabola). The $xy$ term rotates the curve, and $B^2 - 4AC$ is the combination that rotation does not change.
Worked examples
Ejemplo resuelto: An ellipse
Identify $4x^2 + 9y^2 - 16x + 18y - 11 = 0$ and find its center.
- $A = 4$ and $C = 9$ have the same sign and are different: an ellipse.
- Group and factor: $4(x^2 - 4x) + 9(y^2 + 2y) = 11$.
- Complete the squares, adding $4 \cdot 4 = 16$ and $9 \cdot 1 = 9$ to both sides: $4(x - 2)^2 + 9(y + 1)^2 = 36$.
- Divide by $36$: $\frac{(x - 2)^2}{9} + \frac{(y + 1)^2}{4} = 1$.
Respuesta: Ellipse with center $(2, -1)$
Ejemplo resuelto: A hyperbola
Identify $x^2 - y^2 + 4x + 2y - 1 = 0$ and find its center.
- $A = 1$, $C = -1$: opposite signs, a hyperbola.
- $$(x^2 + 4x + 4) - (y^2 - 2y + 1) = 1 + 4 - 1.$$
- $(x + 2)^2 - (y - 1)^2 = 4$.
Respuesta: Hyperbola with center $(-2, 1)$, opening left and right
Ejemplo resuelto: A parabola
Identify $y^2 - 4y - 8x + 20 = 0$ and find its vertex.
- Only $y$ is squared: a parabola.
- $y^2 - 4y + 4 = 8x - 20 + 4$, so $(y - 2)^2 = 8x - 16 = 8(x - 2)$.
Respuesta: Parabola with vertex $(2, 2)$, opening right
Error común
Completing the square inside a factored group but adding the wrong amount to the other side. In $4(x^2 - 4x + 4)$ the $+4$ inside the parentheses is really $+16$, because it is multiplied by $4$. Add $16$ to the other side, not $4$.