Conic Sections
Conic sections are the curves obtained by intersecting a right circular cone with a plane at various angles. Depending on the tilt of the cutting plane relative to the cone's axis, the intersection produces one of four fundamental curves: a circle, an ellipse, a parabola, or a hyperbola. These shapes appear throughout mathematics, physics, astronomy, and engineering — from the orbits of planets to the design of satellite dishes and suspension bridges. The term 'analytic geometry' refers to the study of geometric objects using coordinate systems and algebraic equations. In this framework, every conic section can be described by a second-degree polynomial equation in two variables x and y, of the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0. The values of the coefficients determine which conic is represented, and the discriminant B² − 4AC reveals the type: negative for an ellipse (or circle), zero for a parabola, and positive for a hyperbola. Beyond the standard Cartesian (analytic) form, each conic section can also be expressed parametrically — that is, by writing both x and y as functions of a single parameter t. Parametric forms are especially useful for tracing paths, computing arc lengths, and understanding the geometry of motion along a curve. Together, the analytic and parametric descriptions give a powerful, flexible toolkit for studying these elegant and historically important curves.
- The Circle
- The Ellipse
- The Parabola
- The Hyperbola
- The Discriminant Test
The Circle
A circle with center (h, k) and radius r satisfies (x − h)² + (y − k)² = r². Its parametric form is x = h + r cos(t), y = k + r sin(t), for t ∈ [0, 2π). Every point on the circle is exactly distance r from the center. The circle is the special case of an ellipse where both semi-axes are equal.
The Ellipse
An ellipse centered at the origin with semi-major axis a along the x-axis and semi-minor axis b along the y-axis is described by x²/a² + y²/b² = 1, where a > b > 0. Its parametric form is x = a cos(t), y = b sin(t). The foci are located at (±c, 0) where c² = a² − b². For any point P on the ellipse, the sum of distances from P to the two foci equals 2a — this is the defining focal property of the ellipse.
The Parabola
A parabola with vertex at the origin, opening upward, is given by y = x²/(4p), equivalently x² = 4py, where p is the distance from the vertex to the focus (the focus is at (0, p) and the directrix is y = −p). Its parametric form is x = 2pt, y = pt². For any point P on the parabola, the distance from P to the focus equals its perpendicular distance to the directrix. Horizontal parabolas have the form y² = 4px.
The Hyperbola
A hyperbola centered at the origin with transverse axis along x is given by x²/a² − y²/b² = 1. Its parametric form uses hyperbolic functions: x = a cosh(t), y = b sinh(t) for one branch (or x = ±a sec(t), y = b tan(t) for both branches). The foci are at (±c, 0) where c² = a² + b². For any point P, the absolute difference of distances to the two foci equals 2a. The asymptotes are the lines y = ±(b/a)x.
The Discriminant Test
Given the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, compute Δ = B² − 4AC. If Δ < 0, the curve is an ellipse (or circle when A = C and B = 0). If Δ = 0, it is a parabola. If Δ > 0, it is a hyperbola. Degenerate cases (a point, two lines, or no real curve) can also occur. This algebraic test elegantly encodes the geometric nature of the curve from its equation alone.
The Key Equation
- a — Semi-major axis (length units): Half the length of the longest diameter of the ellipse; a > b > 0.
- b — Semi-minor axis (length units): Half the length of the shortest diameter; determines the 'width' perpendicular to the major axis.
- c — Focal distance (length units): Distance from the center to each focus; satisfies c² = a² − b².
- e — Eccentricity (dimensionless): Ratio e = c/a; ranges from 0 (circle) to values approaching 1 (very elongated ellipse). For a parabola e = 1, for a hyperbola e > 1.
Worked Example
Problem. Find the foci, eccentricity, and parametric equations of the ellipse 9x² + 25y² = 225.
Answer. Foci at (±4, 0), eccentricity e = 4/5, parametric form: x = 5cos(t), y = 3sin(t).
Origins · 1 minApollonius of Perga and how this was discovered
Apollonius wrote the foundational eight-volume treatise 'Conics' (Κωνικά) around 200 BCE, in which he named the parabola, ellipse, and hyperbola and systematically derived their properties from a single cone. His work defined the field for nearly two millennia and was so advanced that it was not significantly surpassed until the development of analytic geometry by Descartes and Fermat in the 17th century. Apollonius earned the nickname 'The Great Geometer' from his contemporaries. His work on conics was so thorough that Kepler, roughly 1800 years later, relied heavily on it when formulating his laws of planetary motion.
Related Concepts
- Vectors in the Plane — Vectors and their dot products are used to derive distance formulas and to analyze the directions of axes and foci in conic sections.
- Parametric Curves — Each conic section has a natural parametric representation; understanding how to work with parametric equations is essential for tracing and analyzing conics.
- Quadratic Forms and Matrices — The general second-degree conic equation Ax²+Bxy+Cy²+...=0 is a quadratic form; matrix methods including eigenvalues reveal the type and orientation of the conic.
- Arc Length of Curves — Computing the perimeter of an ellipse or arc length along any conic requires integration techniques applied to parametric or Cartesian equations.