The Geometric Definitions for Circles and Ellipses

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Gladys Ferguson
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18 Conic Sections Concepts: The Origin of Conic Sections Equations and Graphs of Circles and Ellipses The Geometric Definitions for Circles and Ellipses (Sections 10.1-10.

1 18 Conic Sections Concepts: The Origin of Conic Sections Equations and Graphs of Circles and Ellipses The Geometric Definitions for Circles and Ellipses (Sections ) A conic section or conic is the intersection of a plane and a right circular cone. If the plane does not pass through the vertex of the cone, the conic is either a circle, a parabola, an ellipse, or a hyperbola. See the picture below that was taken from 1

If the plane does pass through the vertex of the cone, then the conic is either a point, a line, or a pair of lines that intersect at the vertex of the cone.

We will be primarily concerned with non-degenerate conics in this class. So far, we have described conics geometrically.

2 If the plane does pass through the vertex of the cone, then the conic is either a point, a line, or a pair of lines that intersect at the vertex of the cone. These three types of conics are called degenerate conics. See the picture below that was taken from We will be primarily concerned with non-degenerate conics in this class. So far, we have described conics geometrically. From an algebraic perspective, all nondegenerate conics have an equation that is equivalent to an equation of the form Ax 2 + Bxy +Cy 2 +Dx+Ey +F = 0 where at least one of A, B, and C is nonzero Circles We already discussed circles earlier this semester. We include a brief reminder here for completeness. Definition 18.1 (Circles) Let P be a point in the plane and r a positive number. The circle with center P and radius r is the set of all points X in the plane such that Distance from X to P equals r. We were able to use this definition and the distance formula to find an equation for a circle with center P(h,k) and radius r. Proposition 18.2 (Equation of a Circle) The circle with center P(h,k) and radius r is the graph of the equation (x h) 2 +(y k) 2 = r 2. 2

3 18.2 Ellipses Definition 18.3 (Ellipses) Let P and Q be points in the plane and r a number that is greater than the distance from P to Q. The ellipse with foci P and Q and constant r is the set of all points X in the plane such that (Distance from X to P) + (Distance from X to Q) = r. A nice demonstration of this definition can be found at: Definition 18.4 The center of an ellipse with foci P and Q is the midpoint of the line segment PQ. The vertices of the ellipse are the two points where the line through P and Q intersects the ellipse. The major axis of the ellipse is the line segment that joins the vertices. The minor axis of the ellipse is the line segment that is perpendicular to the major axis, passes through the center of the ellipse, and connects two points of the ellipse. vertex major axis minor axis center vertex It is possible to use this definition, the distance formula, and some serious algebraic simplification to find an equation for an ellipse with a horizontal or vertical major axis. 3

4 Proposition 18.5 (Standard Equation of an Ellipse with Center at the Origin) Let a > b > 0. x 2 a + y2 2 b = 1 2 is the ellipse with center (0,0), horizontal major axis of length 2a, vertical minor axis of length 2b, and foci (c,0) and ( c,0) where c = a 2 b 2. x 2 b + y2 2 a = 1 2 is the ellipse with center (0,0), vertical major axis of length 2a, horizontal minor axis of length 2b, and foci (0,c) and (0, c) where c = a 2 b 2. Example 18.6 Sketch the graph of 9x 2 +16y 2 = 144. Example 18.7 Sketch the graph of (x 3)2 4 + (y +1)2 25 = 1. 4

5 Proposition 18.8 (Standard Equation of an Ellipse with Center at P(h, k)) Let a > b > 0. (x h) 2 (y k)2 + = 1 a 2 b 2 is the ellipse with center (h,k), horizontal major axis of length 2a, vertical minor axis of length 2b, and foci (c+h,k) and ( c+h,k) where c = a 2 b 2. (x h) 2 (y k)2 + = 1 b 2 a 2 is the ellipse with center (0,0), vertical major axis of length 2a, horizontal minor axis of length 2b, and foci (h,c+k) and (h, c+k) where c = a 2 b Hyperbolas We will not study hyperbolas in depth in this class, but we have included the two basic graphs of a hyperbola and there equations for completeness. y = b a x y = b a x ( a, 0) (a, 0) Equation: x2 a 2 y2 b 2 = 1 5

6 y = a b x (0,a) y = a b x (0, a) Equation: y2 a 2 x2 b 2 = 1 According to your textbook, hyperbolas are used for long-range navigational systems and to describe how light reflects off of telescope and camera lenses. For more information, see the applications in Section 10.2 of your textbook Parabolas We examined parabolas in this class when we studied quadratic functions. Parabolas are much more general than those obtained from quadratic functions. Any graph that can be obtained by a sequence of graph transformations and rotations of the graph of y = x 2 is a parabola. Although we will not study the more general form of a parabola, it interesting to note that the geometric definition of a parabola led to an algorithm that can be used by GPS devices to find the nearest grocery store or, better yet, the nearest ice cream store. If you are interested in this, you should read section 10.3 of your textbook and investigate Voronoi Diagrams and Fortunes Algorithm. 6

This early Greek study was largely concerned with the geometric properties of conics.

This early Greek study was largely concerned with the geometric properties of conics. 4.3. Conics Objectives Recognize the four basic conics: circle, ellipse, parabola, and hyperbola. Recognize, graph, and write equations of parabolas (vertex at origin). Recognize, graph, and write equations