CONICS
Conic sections
are the curves which can be derived from taking slices of a "double-napped" cone. (A double-napped cone, in regular English, is two cones "nose to nose", with the one cone balanced perfectly on the other.) "Section" here is used in a sense similar to that in medicine or science, where a sample (from a biopsy, for instance) is frozen or suffused with a hardening resin, and then extremely thin slices ("sections") are shaved off for viewing under a microscope. If you think of the double-napped cones as being hollow, the curves we refer to as conic sections are what results when you section the cones at various angles.There are plenty of sites and books with pictures illustrating how to obtain the various curves through sectioning, so I won't bore you with more pictures here. And there are books and entire web sites devoted to the history of conics, the derivation and proofs of their formulas, and their various applications. I will not attempt to reproduce that information here.
This lesson, and the conic-specific lessons to which this page links, will instead concentrate on: finding curves, given points and other details; finding points and other details, given curves; and setting up and solving conics equations to solve typical word problems.
There are some basic terms that you should know for this topic:
- center: the point (h, k) at the center of a circle, an ellipse, or an hyperbola.
- vertex (VUR-teks): 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" (VUR-tuh-seez).
- focus (FOH-kuss): 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" (FOH-siy).
- directrix (dih-RECK-triks): a line from which distances are measured in forming a conic; the plural form is "directrices" (dih-RECK-trih-seez).
- axis (AK-siss): a line perpendicular to the directrix passing through the vertex of a parabola; also called the "axis of symmetry"; the plural form is "axes" (ACK-seez).
- 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 (LOH-kuss): 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" (LOH-siy).
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- PREPARED BY: LEVY BANTAYAN
