Conic Sections

A conic section is the intersection of a plane and a right circular cone. By changing the angle of the plane the intersection can be: a circle, an ellipse, a parabola, or a hyperbola. If the plane intersects the vertex of the cone the resulting intersection is a point, line, or intersecting lines (these are called degenerate conics). We are mainly interested in the first four with their center (circle, ellipse and hyperbola) or vertex (parabola) located at the origin. Occasionally we’ll be using conics that have been shifted.


Circle

Ellipse

Parabola

Hyperbola

Point

Line

Lines
 

To see an interactive site where you can rotate the plane go to: http://www.jalacy.com/conics/java.shtml

The ancient Greeks studied conics. Important scientific applications were discovered during the 17th century. Conics also appear in art, architecture and engineering. Wonderful examples can be see at the website: http://ccins.camosun.bc.ca/~jbritton/jbconics.htm (from the Math Forum site: http://mathforum.org/.)

Conics have been defined several ways. One set of definitions involves a set of points in a plane and the distance(s) of these points to points and/or lines.

Circle: the set of points in a plane that are equidistant (the radius r) from a given point (the center (h, k)).

Parabola: the set of points in a plane that are equidistant from a given point (focus F) and a given line (directrix). Standard equation: y = a(x – h) 2 + k. If a > 0, the parabola opens up. If a < 0, the parabola opens down. If x = a(y – k) 2 + h, a > 0 the parabola opens to the right, a < 0, it opens to the left. If the vertex is at (0, 0) the equation is of the form y = ax 2 or x = ay 2.

D: x = -p

Using the above definition of a parabola, let F have coordinates (0, p) and the directrix be x = -p. The distance from P to F equals the distance from P to D. Square both sides: x 2 - 2px + p 2 + y 2 = x 2 + 2px + p 2 which simplifies to y 2 = 4px. This is the equation of the parabola opening to the right with vertex at the origin. The other three orientations of the parabola have a similar form. When the equation of the parabola is in this form it is easy to determine the focus and the directrix since the coefficient of the linear term is 4p where p is the distance from the vertex to the focus and to the directrix.

Ellipse:the set of points in the plane whose distances from two given points (foci F 1, F 2) have a constant sum.
Using this definition and a lot of algebra results in the standard equation for an ellipse centered at the origin. The major axis of this ellipse runs horizontally from one vertex to the other. The minor axis is perpendicular to the major axis and their intersection point is the center. Interchanging x and y in the equation results in an ellipse with a vertical major axis.


(graphic from AMAYTC Summer Institute materials, July 2002 )

Hyperbola: the set of points in the plane whose distances from two given points (foci F 1, F 2) have a constant difference.

As with the ellipse, the definition and algebra result in the standard equation for a hyperbola with center at the origin.

If x and y are interchanged the hyperbola opens along the y-axis.
The website http://www.exploremath.com/activities/activity_list.cfm?categoryID=1 allows you to manipulate the various aspects of a parabola, ellipse or hyperbola.
  One entry found for eccentricity.

Main Entry: ec•cen•tric•i•ty
Pronunciation: "ek-(")sen-'tri-s&-tE
Function: noun
Inflected Form(s): plural -ties
Date: 1545
1 a : the quality or state of being eccentric b : deviation from an established pattern or norm; especially : odd or whimsical behavior
2 a : a mathematical constant that for a given conic section is the ratio of the distances from any point of the conic section to a focus and the corresponding directrix b : the eccentricity of an astronomical orbit used as a measure of its deviation from circularity


 
 
(graphic from AMAYTC Summer Institute materials, July 2002)

(Focus is at (0,0) )
Conics may also be described as plane curves that are the paths (loci) of a point moving such that the ratio of its distance from a fixed point (the focus) to the distance from a fixed line (the directrix) is a constant, called the eccentricity of the curve. As you can see from the definition from the dictionary, eccentricity is important in astronomy. Kepler was the first to propose that planets orbit the sun in ellipses. Newton derived the shape of orbits mathematically using calculus, under the assumption that gravitational force goes as the inverse square of distance, and published his works in “Mathematical Principles of Natural Philosophy”. Depending on the energy of the orbiting body, orbit shapes can be are any of the four types of conic sections are possible.

If the eccentricity is zero, the curve is a circle; if equal to one, a parabola; if less than one, an ellipse; and if greater than one, a hyperbola. (Note: ellipses and hyperbolas have two foci and directrices. In the drawing, the hyperbola is not symmetric to the shown directrix.) Using the above definition, fix a point F(focus) with coordinates (p, 0) and a line D (directrix) in the xy-plane such that D does not go through point P (x, y). Pick a positive number e (which represents the eccentricity of the conic). The conic C will be the locus of all points P in the plane such that d(P,F)/ d(P, D) = e or d(P, F) = e*d(P, D) (#1). Note that C is symmetric about the line through F perpendicular to D. This line is the major axis of the conic. The vertex is the point on the curve which lies on this axis, and in the sketch below, it is at (0, 0). The directrix has equation x = -p/e since the vertex satisfies #1. (d(V, F) = p, ⇒ d(V, D) = p/e.)

d(P, F) =   d(P, D) = x + p/e

It will now be shown that if e = 1, C is a parabola; if e < 1, C is an ellipse; and, if e > 1, C is a hyperbola.
Equation #1 is given algebraically by = e(x + p/e) = ex + p. (#2)
Squaring both sides of #2 gives: x 2 – 2px + p 2 + y 2 = e 2 (x 2 + 2px + p 2).

Case e = 1: gives x 2 – 2px + p 2 + y 2 = x 2 + 2px + p 2 which simplifies to y 2 = 4px. This is the standard form of a parabola with vertex at the origin, opening to the right. It also locates the focus and the directrix of a parabola. Note that this matches the equation found using the previous definition.

Case e ≠ 1: gives x 2 – 2px + p 2 + y 2 = e 2 (x 2 + 2px + p 2) which simplifies to: (1 – e 2)x 2 + y 2 – 2p(1 + e)x = 0. If e < 1, this is an ellipse, and if e > 1, this is a hyperbola. Algebra can be used to write this equation in a form equivalent to the previous ones given for an ellipse and a hyperbola and it can be shown that e = c/a.

For examples see pages A12 – A16 of your text.

Problems

#1 – 8: Identify each equation as the equation of a circle, parabola, ellipse, or hyperbola. If the shape is a parabola or hyperbola, determine which way it opens. If it is an ellipse determine whether the major axis runs along the x, y, or z axis. Give a rough sketch of each curve in the appropriate plane.

Example: y = - x 2 - 1


Answer: curve is a parabola in the xy-plane opening along the negative y-axis.
Example:

Answer: curve is an ellipse in the xz-plane. The major axis runs along the z-axis.
1.    2.   
3.   y 2 = 1 – x 4.   z 2 - x = 1 5.   x 2 = 4 – 4 y 2
6.   y 2 = 9z 2 + 9 7.   x 2 + 4x + z 2 = 0 8.   9x 2 + 7y 2 = 63

#9 – 15: Write an equation for each conic. Each parabola has its vertex at the origin, and each ellipse or hyperbola is centered at the origin. You can assume that each conic is in the xy-plane.

Example: Focus (-1, 0): e = ¼
Answer: Since e < 1, you know this is an ellipse. The center is at (0, 0) and a focus is at (-1, 0) ⇒ c = 1.

e = c/a ⇒ a = 4. For an ellipse, a 2 – b 2 = c 2 ⇒ b 2 = 15. ∴the equation is:
9.   Focus (0, 8); e = 110.   Focus (0, -2); e = ½11.   Vertex (0, -6); e = 2
12.   Vertex (4, 0); e = 5/3 13.   Focus (0, 2); e = 6/5 
14.   Vertical major axis of length 6; e = 4/515.   x-intercepts –4 and 4; e = 7/3