Conic form refers to the general second-degree equation that describes every conic section: Ax² + Bxy + Cy² + Dx + Ey + F = 0. This single equation, depending on the values of its coefficients, can produce a circle, ellipse, parabola, or hyperbola. These four curves are called “conic sections” because each one results from slicing a cone at a different angle, and the equation above is the algebraic way to represent all of them at once.
The General Equation
The general conic form uses six coefficients (A through F) applied to two variables, x and y. The squared terms (Ax² and Cy²) control whether the curve is open or closed. The Bxy term, called the cross term, indicates whether the conic has been rotated away from the standard horizontal and vertical axes. D and E shift the curve’s position, and F is a constant that adjusts the overall equation.
Most introductory math courses work with conics where B = 0, meaning the curve sits neatly along the x and y axes. When B is not zero, the conic is tilted, and you need a rotation formula to eliminate that cross term before you can identify the shape.
How to Identify the Conic Type
You can classify any conic from the general equation using the discriminant: B² − 4AC. This single value tells you which shape the equation describes.
- Parabola: B² − 4AC = 0
- Ellipse: B² − 4AC is negative
- Circle: B² − 4AC is negative, and A = C, and B = 0
- Hyperbola: B² − 4AC is positive
A circle is technically a special case of an ellipse where the two axes have equal length. That’s why the discriminant conditions overlap: a circle meets all the ellipse criteria plus the additional requirement that A equals C with no cross term.
Standard Forms for Each Conic
While the general equation is useful for classification, each conic type has its own “standard form” that reveals key geometric features at a glance. These standard forms assume the conic is centered at (or has its vertex at) a point (h, k).
Circle
(x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. This is the simplest conic. Both squared terms have the same coefficient, and there’s no xy term.
Ellipse
((x − h)² / a²) + ((y − k)² / b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes. When a is larger, the ellipse is wider than it is tall. Swap a and b between the x and y terms, and the ellipse stretches vertically instead.
Parabola
(x − h)² = 4p(y − k) for a parabola that opens up or down, or (y − k)² = 4p(x − h) for one that opens left or right. The vertex sits at (h, k), and the value p tells you how far the focus is from the vertex. A larger p produces a wider, shallower curve.
Hyperbola
((x − h)² / a²) − ((y − k)² / b²) = 1 for a hyperbola that opens left and right. Reversing which term is subtracted flips it to open up and down. Unlike the ellipse, the subtraction sign means the curve has two separate branches that never connect.
Eccentricity: One Number for Shape
Eccentricity (written as “e”) measures how much a conic deviates from being a perfect circle. It gives you a single number that captures the shape’s character.
- Circle: e = 0
- Ellipse: 0 < e < 1
- Parabola: e = 1
- Hyperbola: e > 1
An ellipse with eccentricity close to 0 looks nearly circular, while one approaching 1 looks long and narrow. A parabola sits right at the boundary, and hyperbolas with higher eccentricity have branches that spread apart more sharply. This concept connects directly to the geometric definition of a conic: every conic section is the set of all points where the ratio of the distance to a fixed point (the focus) and the distance to a fixed line (the directrix) equals the eccentricity. When that ratio is exactly 1, you get a parabola. Below 1, the points stay closer to the focus than the directrix, forming a closed ellipse. Above 1, the points spread outward into a hyperbola’s two branches.
What the Cross Term Does
The Bxy term in the general equation is what makes many students stumble. When B is zero, the conic’s axes line up with the coordinate axes, and you can convert to standard form by completing the square. When B is not zero, the conic is rotated, and the equation doesn’t match any of the clean standard forms above.
To handle this, you calculate a rotation angle that eliminates the cross term, transforming the equation into a new coordinate system where the conic sits in standard position. The key insight is that rotating the axes doesn’t change the shape of the conic. It only changes how it’s oriented relative to your coordinate grid. The discriminant B² − 4AC stays the same regardless of rotation, which is why it works as a reliable classifier even when the equation looks messy.
Degenerate Cases
Not every second-degree equation produces a recognizable curve. When the equation’s coefficients combine in certain ways, you get what’s called a degenerate conic. These occur geometrically when the slicing plane passes directly through the tip of the cone instead of cutting through its sides. The three degenerate cases are a single point (like 4x² + 4y² = 0, which is satisfied only at the origin), a single line, or two intersecting lines.
You can think of these as edge cases. A degenerate “ellipse” collapses to a point. A degenerate “parabola” becomes a line. A degenerate “hyperbola” becomes two lines crossing each other. The general equation still technically describes them, but they’re not the curves you typically care about.
Where Conic Forms Appear in Practice
Conic sections aren’t just textbook geometry. Johannes Kepler proved that planets orbit the sun in ellipses, not perfect circles, and astronomers still use the mathematical properties of ellipses to predict planetary orbits and the arrival of comets. The closer a planet’s orbital eccentricity is to zero, the more circular its path. Earth’s orbit has an eccentricity of about 0.017, which is why it’s nearly circular but not quite.
Parabolas show up in satellite dishes and radio telescopes. The dish surface is a paraboloid (a parabola rotated in three dimensions), and the actual receiver sits at the focus point. Any signal arriving parallel to the dish’s axis reflects off the surface and converges at that single focus, which is why parabolic shapes are so effective at collecting weak signals. The same principle works in reverse for flashlights and car headlights: a bulb at the focus sends light outward in a parallel beam.
Hyperbolas appear in navigation systems and in the geometry of sonic booms, where the shock wave traces a cone whose cross section is a hyperbola. Even the cooling towers at power plants use a hyperbolic profile because the shape provides structural strength with minimal material.