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.
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
Parabola: the set of points in a plane that are equidistant from a given point (focus F) and a given line (directrix). Standard equation:
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:
Ellipse:the set of points in the plane whose distances from two given points
|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.|
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.
|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
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
Squaring both sides of #2 gives:
Case e = 1: gives
Case e ≠ 1: gives
For examples see pages A12 – A16 of your text.
#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.
Answer: curve is a parabola in the xy-plane opening along the negative
Answer: curve is an ellipse in the xz-plane. The major axis runs along the
#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
e = c/a ⇒ a = 4. For an ellipse, a 2 – b 2 = c 2 ⇒ b 2 = 15. ∴the equation is: