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Conic Sections

What is it?

A conic section is a curve obtained as the intersection of a cone with a plane. A conic consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity.

Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix definition of a conic the circle is a limiting case with eccentricity 0 (wikipedia) cone-cir.jpg cone-ell.jpg cone-hyp.jpg cone-par.jpg
Images from math2.org/math/algebra/conics.htm

The placement of the two cones "nose to nose", with the one cone balanced perfectly on the other is referred to as double-napped.



Vocabulary

  • center: the point (h, k) at the center of a circle, an ellipse, or an hyperbola.

  • vertex: in the case of a parabola, the point (h, k) at the end of a parabola; in the case of an ellipse, an end of the major axis; in the case of an hyperbola, the turning point of a branch of an hyperbola; the plural form is vertices.

  • focus: a point from which distances are measured in forming a conic; a point at which these distance-lines converge, or focus; the plural form is foci.

  • directrix: a line from which distances are measured in forming a conic; the plural form is directrices .

  • axis: a line perpendicular to the directrix passing through the vertex of a parabola; also called the axis of symmetry; the plural form is axes .

  • major axis: a line segment perpendicular to the directrix of an ellipse and passing through the foci; the line segment terminates on the ellipse at either end; also called the principal axis of symmetry; the half of the major axis between the center and the vertex is the semi-major axis.

  • minor axis: a line segment perpendicular to and bisecting the major axis of an ellipse; the segment terminates on the ellipse at either end; the half of the minor axis between the center and the ellipse is the semi-minor axis.

  • locus: a set of points satisfying some condition or set of conditions; each of the conics is a locus of points that obeys some sort of rule or rules; the plural form is loci.

Identification

Conics have typical shapes and typical forms of their equations:
conic02.gif conic01.gif conic07.gif
conic03.gif conic05.gif conic06.gif

parabola: Ax2 + Dx + Ey = 0
circle: x2 + y2 + Dx + Ey + F = 0
ellipse: Ax2 + Cy2 + Dx + Ey + F = 0
hyperbola: Ax2 – Cy2 + Dx + Ey + F = 0

This is the sequence of tests you should keep in mind:

  1. Are both variables squared?
    No: It's a parabola. (6x2 + 12x – y + 15 = 0)
    Yes: Go to the next test....

  2. Do the squared terms have opposite signs?
    Yes: It's an hyperbola.(x2 – y2 + 3x – 2y – 43 = 0)
    No: Go to the next test....

  3. Are the squared terms multiplied by the same number?
    Yes: It's a circle. (3x2 + 3y2 – 6x + 9y – 14 = 0)
    No: It's an ellipse.(x2 + 2y2 + 4x + 2y – 27 = 0)