Second-Degree Equations: Classifying Conic Sections Simply

The equation ax² + 2hxy + by² + 2gx + 2fy + c = 0 is called the general second-degree equation, and it is the single formula that produces every conic section: circles, ellipses, parabolas, and hyperbolas. The letters a, h, b, g, f, c are just numbers, and which curve you get depends only on how those numbers combine.

Classifying matters because it lets you name the curve without plotting a single point. A physicist who sees the equation of a comet's path can tell instantly whether it will return (ellipse), visit once (parabola), or fly past forever (hyperbola). Engineers, graphics programmers, and astronomers all use this quick test regularly.

Key Concepts

• Discriminant Δ = abc + 2fgh − af² − bg² − ch² decides whether the curve is genuine or degenerate.
• Δ = 0 means degenerate: a pair of straight lines, a single point, or parallel lines.
• Δ ≠ 0 gives a proper conic, named by h² − ab: negative means ellipse, zero means parabola, positive means hyperbola.
• Circle: the special ellipse with a = b and h = 0, meaning no xy term and equal x² and y² coefficients.
• The 2hxy term rotates the curve; h² − ab measures whether that rotation overpowers the curvature into a hyperbola or stays bound as an ellipse.
• Order of testing: always compute Δ first, because a degenerate case can mimic any conic's h² − ab sign.

Formulas

• General equation: ax² + 2hxy + by² + 2gx + 2fy + c = 0.
• Discriminant: Δ = abc + 2fgh − af² − bg² − ch².
• Classification: Δ = 0 means degenerate; Δ ≠ 0 with h² − ab < 0 gives an ellipse, h² − ab = 0 gives a parabola, h² − ab > 0 gives a hyperbola; a = b and h = 0 gives a circle.
• Assumptions: a, h, b are not all zero (otherwise the equation is not second degree); coefficients are real numbers.
• Units: pure numbers in the test itself; in applications x and y carry their own units.
• Limits: the test names the shape but does not give centre, axes, or orientation; those need completing the square or rotation.

Worked Example

Classify x² + 4xy + 4y² + 2x + 6y + 1 = 0. Step 1: read off a = 1, 2h = 4 so h = 2, b = 4, 2g = 2 so g = 1, 2f = 6 so f = 3, c = 1. Step 2: Δ = abc + 2fgh − af² − bg² − ch² = (1)(4)(1) + 2(3)(1)(2) − (1)(9) − (4)(1) − (1)(4) = 4 + 12 − 9 − 4 − 4 = −1. Step 3: Δ ≠ 0, so it is a genuine conic. Step 4: h² − ab = 4 − 4 = 0, so it is a parabola (a rotated one, because of the xy term). Bonus check: x² + y² − 4x + 6y + 4 = 0 gives a = b = 1, h = 0, g = −2, f = 3, c = 4; Δ = 4 − 9 − 4 = −9 ≠ 0; h² − ab = −1 < 0 with a = b and h = 0, so it is a circle with centre (2, −3) and radius √(4 + 9 − 4) = 3.

Common Misconceptions

• "h² − ab < 0 always means ellipse." Only when Δ ≠ 0 is checked first; with Δ = 0 the same sign can hide an imaginary pair of lines or a single point.
• "Any equation with x² and y² terms is a circle." A circle needs equal coefficients on x² and y², no xy term, and Δ ≠ 0; without all three it is an ellipse or something else.
• "A parabola must look like y = x²." A rotated parabola with an xy term, like the worked example, is still a parabola; h² − ab = 0 is the real test, not the orientation.

Real-World Applications

• Orbital mechanics: the same test separates bound elliptical orbits from one-visit parabolic and hyperbolic trajectories of comets and spacecraft.
• Architecture: parabolic arches and hyperbolic cooling towers are chosen by exactly these curve properties.
• Computer graphics: rendering and detecting conic curves in design software starts from this classification.
• Optics: the shapes of mirrors and lenses, and the paths of light through them, are conic sections identified this way.

Compute Δ = abc + 2fgh − af² − bg² − ch²: zero means degenerate, non-zero sends you to h² − ab, whose sign names the conic, negative for ellipse, zero for parabola, positive for hyperbola, with a = b and h = 0 singling out the circle.