Conic sections

Builds on

Required prerequisites, all the way down: 26

You will learn to

• Describe each conic as a section of a double cone and as a locus defined by distances to foci or to a focus and directrix.
• Convert between geometric descriptions and standard equations, completing the square where necessary.
• Explain and use the reflective property of the parabola and of the ellipse.

Definition

• An ellipse is the set of points whose distances to two fixed points (the foci) have a constant sum greater than the distance between the foci.
• A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix) not through it.
• A hyperbola is the set of points whose distances to two foci have a constant difference in absolute value, smaller than the distance between the foci.
• The eccentricity e of a non-circular conic is the constant ratio of the distance to a focus and the distance to the corresponding directrix.

Theorem

• Every non-degenerate curve given by a second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 is an ellipse, parabola or hyperbola, classified (up to degenerate cases) by the sign of B² − 4AC.
• A ray parallel to the axis of a parabola reflects off the parabola through the focus.
• A conic is an ellipse, parabola or hyperbola according as e < 1, e = 1 or e > 1.

Common misconceptions

• Thinking a parabola is half of a hyperbola, or that it eventually becomes straight; a parabola has no asymptotes.
• Believing every second-degree equation describes a curve; some describe a point, a pair of lines or nothing at all.

Leads to

Optional discoveries

• Projective geometry In the projective plane ellipses, parabolas and hyperbolas become a single kind of curve: they differ only in how they meet the line at infinity.
• Classical mechanics Kepler's first law says planets move on ellipses with the Sun at a focus; Newton derived it from the inverse-square law of gravitation.Wonder Trail →

Further reading

• Keith Kendig, Conics (2005)
• Tom M. Apostol, Calculus, Volume 1 (1967)