Enter coefficients for Ax² + Bxy + Cy² + Dx + Ey + F = 0. If B is nonzero, the calculator rotates axes before completing the square.
Conic Section Classifier
Use this Conic Section Classifier to enter values, adjust options, and review results in a compact responsive workspace.
Results are calculated automatically as you enter data.
Classification
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What Is Classifying Conic Sections from a General Equation?
A conic section is a curve that can appear when a plane cuts through a cone. In coordinate geometry, the same family of curves also appears as the graph of a second-degree equation in two variables:
$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
$$
Depending on the coefficients, this equation can represent a circle, ellipse, parabola, hyperbola, a degenerate case such as a point or line, or sometimes no real graph at all.
Classifying a conic means deciding which kind of curve the equation represents. Rewriting it means changing the equation into a more recognizable standard form. Standard form makes the shape easier to understand because it reveals features such as the center, vertex, direction of opening, and whether the axes have been rotated.
This matters because the general equation does not always look like the familiar textbook forms. For example, an equation with an \(xy\) term may describe an ordinary ellipse or hyperbola whose axes are tilted relative to the usual \(x\)- and \(y\)-axes. Classification and rewriting help connect the algebraic equation to the actual graph.
Why Conic Classification Matters
Conic sections appear throughout algebra, precalculus, calculus, analytic geometry, physics, optics, and engineering. Students often meet them first as separate standard forms, but many real problems produce equations in expanded general form.
Classifying and rewriting a conic helps you:
- recognize the curve before graphing it;
- find the center or vertex more reliably;
- detect when a graph is rotated;
- distinguish a true curve from a degenerate case;
- check whether an equation has no real points;
- compare different-looking equations that describe the same curve.
The process is also a useful algebra skill. It combines discriminants, rotation of axes, completing the square, and careful interpretation of signs.
Key Terms to Know
- Conic section: A circle, ellipse, parabola, or hyperbola, along with related degenerate cases.
- General second-degree equation: An equation of the form \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).
- Coefficient: A number multiplying a term. In the general conic equation, \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) are coefficients.
- Cross-term: The \(Bxy\) term. When \(B \ne 0\), the conic is usually rotated relative to the original axes.
- Discriminant: The expression \(B^2 - 4AC\), used as an important clue for the conic type.
- Rotation of axes: A change from the original \(x\)-\(y\) axes to tilted axes, often called \(u\)-\(v\) or \(x'\)-\(y'\), to remove the \(xy\) term.
- Completing the square: An algebra method for rewriting quadratic terms into squared binomials, such as \(x^2 - 4x = (x - 2)^2 - 4\).
- Standard form: A rewritten equation that clearly shows the type and key features of a conic.
- Degenerate conic: A limiting case such as a point, a line, a pair of lines, or another non-standard result.
- No real graph: A result where no real coordinate pair satisfies the equation, such as \(x^2 + y^2 + 1 = 0\).
How Classifying and Rewriting Conic Sections Works
The starting point is the general equation:
$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
$$
The quadratic part is:
$$
Ax^2 + Bxy + Cy^2
$$
The signs and relationships among \(A\), \(B\), and \(C\) give a first clue about the curve. The most common shortcut is the discriminant:
$$
\Delta = B^2 - 4AC
$$
For nondegenerate real conics, the sign of \(\Delta\) usually points to the conic family:
| Discriminant | Main classification clue |
|---|---|
| \(\Delta < 0\) | Ellipse-type curve, with a circle as a special case |
| \(\Delta = 0\) | Parabola-type curve |
| \(\Delta > 0\) | Hyperbola-type curve |
The word “type” is important. The discriminant tells you the broad family, but it does not by itself guarantee that the equation has a normal visible curve. Constants and linear terms can still produce a point, a line, a pair of lines, or no real graph.
Removing the \(xy\) Term
If \(B = 0\), the conic is already aligned with the original coordinate axes. If \(B \ne 0\), the \(xy\) term usually means the conic is rotated. A rotation of axes can remove that cross-term.
One common computational form for the rotation angle is:
$$
\theta = \frac{1}{2}\operatorname{atan2}(B, A - C)
$$
After rotating the coordinate system, the equation can be rewritten in variables such as \(u\) and \(v\):
$$
A'u^2 + C'v^2 + D'u + E'v + F' = 0
$$
The prime marks show that these are the coefficients after rotation. The important point is that the \(uv\) cross-term has been removed, so the equation is easier to recognize.
Completing the Square
Once the cross-term is gone, completing the square turns the equation into standard form. If both squared terms are present, the equation can often be arranged like this:
$$
A'(u - u_0)^2 + C'(v - v_0)^2 = R
$$
The signs of \(A'\), \(C'\), and \(R\) then reveal the shape:
- Same-sign squared terms with a positive right-hand side usually indicate an ellipse or circle.
- Opposite-sign squared terms usually indicate a hyperbola.
- One squared term and one remaining linear term usually indicate a parabola.
- Same-sign squared terms with a zero or impossible right-hand side may indicate a point or no real graph.
For parabolas, the standard form often looks like one of these:
$$
(u - h)^2 = 4p(v - k)
$$
or
$$
(v - k)^2 = 4p(u - h)
$$
The vertex is \((h, k)\) in the rotated coordinate system, and the sign of \(p\) determines the direction of opening.
The Role of the Determinant
A conic can also be represented with a symmetric matrix. In that form, the determinant of the full coefficient matrix helps identify degenerate behavior:
$$
\begin{vmatrix}
A & \frac{B}{2} & \frac{D}{2} \\
\frac{B}{2} & C & \frac{E}{2} \\
\frac{D}{2} & \frac{E}{2} & F
\end{vmatrix}
$$
A value near zero is a warning sign that the equation may collapse into a degenerate case. Because numerical calculations use rounding and tolerances, a result that is extremely close to zero should be interpreted carefully.
Examples of Conic Classification in Practice
Example 1: A Circle from General Form
Consider:
$$
x^2 + y^2 - 4x + 6y - 12 = 0
$$
The coefficients are:
$$
A = 1,\quad B = 0,\quad C = 1,\quad D = -4,\quad E = 6,\quad F = -12
$$
The discriminant is:
$$
\Delta = B^2 - 4AC = 0^2 - 4(1)(1) = -4
$$
A negative discriminant suggests an ellipse-type curve. Since \(A = C\) and \(B = 0\), this may be a circle if the completed-square form has a positive radius squared.
Complete the square:
$$
x^2 - 4x + y^2 + 6y = 12
$$
$$
(x - 2)^2 - 4 + (y + 3)^2 - 9 = 12
$$
$$
(x - 2)^2 + (y + 3)^2 = 25
$$
So the graph is a circle with center \((2, -3)\) and radius \(5\).
Example 2: A Parabola from General Form
Consider:
$$
x^2 - 4x - 8y + 12 = 0
$$
Here:
$$
A = 1,\quad B = 0,\quad C = 0
$$
The discriminant is:
$$
\Delta = 0^2 - 4(1)(0) = 0
$$
A zero discriminant suggests a parabola-type curve. Complete the square in \(x\):
$$
x^2 - 4x - 8y + 12 = 0
$$
$$
(x - 2)^2 - 4 - 8y + 12 = 0
$$
$$
(x - 2)^2 = 8(y - 1)
$$
This is a parabola with vertex \((2, 1)\). It opens upward because the right side is a positive multiple of \(y - 1\).
Example 3: A Rotated Ellipse
Consider:
$$
5x^2 + 6xy + 5y^2 - 16x - 8y + 12 = 0
$$
The discriminant is:
$$
\Delta = 6^2 - 4(5)(5) = 36 - 100 = -64
$$
The negative value suggests an ellipse-type curve, but the \(6xy\) term shows that the axes are rotated. Since \(A = C\), the rotation angle that removes the cross-term is \(45^\circ\) using the usual positive rotation convention.
With rotated coordinates \(u\) and \(v\), the equation becomes:
$$
8u^2 + 2v^2 - 12\sqrt{2}u + 4\sqrt{2}v + 12 = 0
$$
Complete the square:
$$
8\left(u - \frac{3\sqrt{2}}{4}\right)^2 + 2(v + \sqrt{2})^2 = 1
$$
In standard form:
$$
\frac{\left(u - \frac{3\sqrt{2}}{4}\right)^2}{\frac{1}{8}} + \frac{(v + \sqrt{2})^2}{\frac{1}{2}} = 1
$$
This is an ellipse, but its standard form is written in rotated coordinates rather than the original \(x\)-\(y\) coordinates.
Example 4: A No-Real-Graph Edge Case
Consider:
$$
x^2 + y^2 + 1 = 0
$$
The discriminant is:
$$
\Delta = 0^2 - 4(1)(1) = -4
$$
That is ellipse-type by discriminant, but the equation can be rewritten as:
$$
x^2 + y^2 = -1
$$
No real numbers \(x\) and \(y\) can make the sum of two squares equal \(-1\). So this equation has no real graph, even though its discriminant is negative.
How to Interpret the Result
A conic classification result tells you what kind of real graph the equation appears to represent after rotation, completing the square, and checking special cases.
- Circle: A special ellipse with equal squared-term coefficients and no rotation. Its standard form usually looks like \((x - h)^2 + (y - k)^2 = r^2\).
- Ellipse: A closed oval-type curve. In standard form, two squared terms with the same sign are set equal to a positive value.
- Parabola: A curve with one squared direction and one linear direction. It has a vertex rather than a center.
- Hyperbola: A two-branch curve. In standard form, one squared term is subtracted from another.
- Degenerate conic: The equation collapses into something like a point, line, pair of lines, or double line.
- No real graph: The equation has no real coordinate solutions.
- Entire plane: If every coefficient is zero, the equation is \(0 = 0\), which is true for every point in the plane.
When the standard form uses \(u\) and \(v\), those are rotated coordinates. The center or vertex is still meaningful, but it is being described after changing axes. Do not assume that a \(u\)-axis or \(v\)-axis is the same as the original \(x\)-axis or \(y\)-axis.
The graph is best treated as a visual aid. For exact work, rely on the classification, equations, and algebraic steps rather than only the plotted curve.
Common Mistakes and Misconceptions
One common mistake is entering an equation before moving every term to one side. The coefficient form assumes the equation equals zero:
$$
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
$$
For example, if the equation is \(x^2 + y^2 = 25\), enter it as:
$$
x^2 + y^2 - 25 = 0
$$
Another common mistake is putting the \(xy\) coefficient in the wrong place. In \(6xy\), the coefficient \(6\) belongs to \(B\), not to \(A\) or \(C\).
Users also sometimes treat fractions such as "1/2" as numeric input. If the input fields require finite decimal numbers, use "0.5" instead.
A negative discriminant does not always mean a visible ellipse. It means the equation is ellipse-type before checking whether the right-hand side of the completed-square form is positive, zero, or impossible over the real numbers.
Finally, some textbooks use \(4AC - B^2\) instead of \(B^2 - 4AC\) as the discriminant. That convention reverses the signs in the classification table, so always check which definition is being used.
When to Use Conic Classification
Use conic classification when you have a second-degree equation in \(x\) and \(y\) and need to understand its graph. It is especially useful when:
- the equation is expanded rather than written in standard form;
- the equation contains an \(xy\) term;
- you need the center, vertex, or rotated standard form;
- you want to check whether a graph is a circle, ellipse, parabola, or hyperbola;
- you suspect the equation may be degenerate;
- you need to compare a plotted graph with an algebraic result.
This process is most useful for real two-dimensional Cartesian coordinate graphs. It is not meant for complex-coordinate geometry, symbolic coefficient analysis, or three-dimensional quadric surfaces.
Limitations and Things to Keep in Mind
Conic classification is powerful, but it depends on accurate coefficients. A small typing error can change the discriminant, the rotation angle, or the final type.
Numerical tools also use scale-relative tolerances. This calculator bases near-zero coefficient checks on a small multiple of Number.EPSILON times the relevant coefficient scale, so ordinary rescaling does not change the decision. Borderline cases should still be checked carefully if exact classification matters.
Displayed numbers may be rounded. The invariant table uses scale-normalized discriminant and determinant values so very large or tiny common coefficient scales do not turn nonzero values into misleading zero or undefined displays. If you are doing exact symbolic work, keep the original fractions or radicals in your own calculations.
A graphing preview is approximate. If the graph is drawn from sampled points in a finite window, a branch of a hyperbola, a far-away vertex, or a very narrow feature may be outside the visible region.
Degenerate conics can be subtle. A result such as a line, double line, point, pair of lines, or no real graph may need additional algebra if you need an exact factorization in the original \(x\)-\(y\) coordinates.
How to Use This Calculator
- Rewrite your equation so that all terms are on one side and the other side is zero.
- Enter finite numeric values for \(A\), \(B\), \(C\), \(D\), \(E\), and \(F\) in \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\).
- Use an example preset if you want to load a sample circle, ellipse, parabola, hyperbola, rotated conic, or degenerate conic.
- Review the classification result, status message, center or vertex, standard form, and graph note.
- Check the invariants and calculation steps to see the discriminant, determinant, rotation angle, rotated equation, and warnings.
- Interpret any standard form in \(u\)-\(v\) coordinates as a rotated-axis form when a rotation is shown.
Frequently Asked Questions
What does the discriminant \(B^2 - 4AC\) tell me?
It gives a quick classification clue. For nondegenerate real conics, a negative value points to an ellipse-type curve, zero points to a parabola-type curve, and a positive value points to a hyperbola-type curve. You still need to check degenerate and no-real-graph cases.
Why does an \(xy\) term mean the conic may be rotated?
The familiar standard forms for circles, ellipses, parabolas, and hyperbolas do not contain an \(xy\) term when their axes line up with the coordinate axes. An \(xy\) term usually appears when the curve has been tilted. Rotating the coordinate axes can remove that term and make the standard form easier to see.
Is a circle an ellipse?
In analytic geometry, a circle is commonly treated as a special case of an ellipse where the two squared directions have the same scale. In coefficient terms, a circle in non-rotated coordinates has equal \(x^2\) and \(y^2\) coefficients and no \(xy\) term.
Why can the result say “no real graph” even when the discriminant points to an ellipse?
The discriminant only looks at the quadratic part of the equation. After completing the square, the equation may require a sum of squares to equal a negative number, which is impossible for real coordinates. In that case, there is no real graph.
What is a degenerate conic?
A degenerate conic is a limiting case where the equation does not produce a usual curved conic. Common examples include a single point, a line, two intersecting lines, two parallel lines, or a double line. These cases still come from second-degree equations, but their graphs are not ordinary ellipses, parabolas, or hyperbolas.
Why is the standard form sometimes written with \(u\) and \(v\) instead of \(x\) and \(y\)?
The variables \(u\) and \(v\) represent rotated axes. They are used when the original equation has an \(xy\) term and the curve is easier to describe after a rotation. The shape is the same curve, but it is being viewed from a tilted coordinate system.
Sources and References
Books and Textbooks
- Jay Abramson. College Algebra 2e. OpenStax, 2021. Chapter 8, “Analytic Geometry,” especially Sections 8.1 “The Ellipse,” 8.2 “The Hyperbola,” 8.3 “The Parabola,” and 8.4 “Rotation of Axes.” https://openstax.org/books/college-algebra-2e/pages/8-introduction-to-analytic-geometry
- Gilbert Strang and Edwin “Jed” Herman. Calculus Volume 2. OpenStax, 2016. Section 7.5 “Conic Sections,” including standard forms, eccentricity, and general second-degree equations. https://openstax.org/books/calculus-volume-2/pages/7-5-conic-sections
Online and Supplemental Sources
- Eric W. Weisstein. “Conic Section.” MathWorld—A Wolfram Web Resource. Accessed June 28, 2026. https://mathworld.wolfram.com/ConicSection.html
- Wikimedia Foundation. “Matrix representation of conic sections.” Wikipedia. Last edited May 2, 2026. Accessed June 28, 2026. Used as a supplemental reference for the matrix form and determinant-based degeneracy check. https://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections