Chapter 9 Conic Sections and Analytic Geometry 9.1 The Ellipse Copyright 2014, 2010, 2007 Pearson Education, Inc. 1
Objectives: Graph ellipses centered at the origin. Write equations of ellipses in standard form. Graph ellipses not centered at the origin. Solve applied problems involving ellipses. Copyright 2014, 2010, 2007 Pearson Education, Inc. 2
Definition of an Ellipse An ellipse is the set of all points, P, in a plane the sum of whose distances from two fixed points, F 1 and F 2, is constant. These two fixed points are called the foci (plural of focus). The midpoint of the segment connecting the foci is the center of the ellipse. Copyright 2014, 2010, 2007 Pearson Education, Inc. 3
Horizontal and Vertical Elongation of an Ellipse An ellipse can be elongated in any direction. The line through the foci intersects the ellipse at two points, called the vertices (singular: vertex). The line segment that joins the vertices is the major axis. The midpoint of the major axis is the center of the ellipse. The line segment whose endpoints are on the ellipse and that is perpendicular to the major axis at the center is called the minor axis of the ellipse. Copyright 2014, 2010, 2007 Pearson Education, Inc. 4
Standard Form of the Equations of an Ellipse The standard form of the equation of an ellipse with center at the origin, and major and minor axes of lengths 2a and 2b (where a and b are positive, and a 2 > b 2 ) is 1 or 1. a b b a Copyright 2014, 2010, 2007 Pearson Education, Inc. 5
Standard Form of the Equations of an Ellipse (continued) The vertices are on the major axis, a units from the center. The foci are on the major axis, c units from the center. For both equations, b 2 = a 2 c 2. Equivalently, c 2 = a 2 b 2. Copyright 2014, 2010, 2007 Pearson Education, Inc. 6
Example: Graphing an Ellipse Centered at the Origin Graph and locate the foci: Express the equation in standard form. 16x 9y 144 144 144 144 Find the vertices. Because the denominator of y 2 is greater than the denominator of x 2, the major axis is vertical. 2 a 16 a 4 The vertices are (0, 4) and (0, 4). 16x 9y 144. 9 16 1 Copyright 2014, 2010, 2007 Pearson Education, Inc. 7
Example: Graphing an Ellipse Centered at the Origin (continued) Graph and locate the foci: 1. 9 16 Find the endpoints of the (horizontal) minor axis. 2 b 9 b 3 The endpoints of the minor axis are (0, 3) and (0, 3). Find the foci. 2 c a b 2 c 16 9 7 The foci are c 7 0, 7 and 0, 7. Copyright 2014, 2010, 2007 Pearson Education, Inc. 8
Example: Graphing an Ellipse Centered at the Origin Graph and locate the foci: 1. 9 16 The major axis is vertical. The vertices are (0, 4) and (0, 4). The endpoints of the minor axis are ( 3, 0) and (3, 0). The foci are 0, 7 and 0, 7. Copyright 2014, 2010, 2007 Pearson Education, Inc. 9
Example: Finding the Equation of an Ellipse from Its Foci and Vertices Find the standard form of the equation of an ellipse with foci at ( 2, 0) and (2, 0) and vertices at ( 3, 0) and (3, 0). Because the foci, ( 2, 0) or (2, 0), are located on the x- axis, the major axis is horizontal. The center of the ellipse is midway between the foci, located at (0, 0). Thus the form of the equation is 1. a b The distance from the center, (0, 0) to either vertex, ( 3, 0) or (3, 0), is 3. Thus a = 3. Copyright 2014, 2010, 2007 Pearson Education, Inc. 10
Example: Finding the Equation of an Ellipse from Its Foci and Vertices (continued) Find the standard form of the equation of an ellipse with foci at ( 2, 0) and (2, 0) and vertices at ( 3, 0) and (3, 0). a = 2, we must find b 2. The distance from the center, (0, 0) to either focus, ( 2, 0) or (2, 0), is 2, so c = 2. c a b b a c 2 b 3 2 9 4 5 The equation is 9 5 1. Copyright 2014, 2010, 2007 Pearson Education, Inc. 11
Standard Forms of Equations of Ellipses Centered at (h, k) Copyright 2014, 2010, 2007 Pearson Education, Inc. 12
Example: Graphing an Ellipse Centered at (h, k) Graph: ( x1) ( y2) 9 4 The form of the equation is 1. ( x h) ( y k) a b 1 h = 1, k = 2. Thus, the center is ( 1, 2). a 2 = 9, b 2 = 4. a 2 > b 2, the major axis is horizontal and parallel to the x-axis. Copyright 2014, 2010, 2007 Pearson Education, Inc. 13
Example: Graphing an Ellipse Centered at (h, k) (continued) Graph: ( x1) ( y2) 9 4 1. The center is at ( 1, 2). The endpoints of the major axis (the vertices) are 3 units right and 3 units left from center. 3 units right ( 1 + 3, 2) = (2, 2) 3 units left ( 1 3, 2) = ( 4, 2) The vertices are (2, 2) and ( 4, 2). Copyright 2014, 2010, 2007 Pearson Education, Inc. 14
Example: Graphing an Ellipse Centered at (h, k) (continued) Graph: ( x1) ( y2) 9 4 1. The center is at ( 1, 2). The endpoints of the minor axis are 2 units up and 2 units down from the center. 2 units up ( 1, 2 + 2) = ( 1, 4) 2 units down ( 1, 2 2) = ( 1, 0) The endpoints of the minor axis are ( 1, 4) and ( 1, 0). Copyright 2014, 2010, 2007 Pearson Education, Inc. 15
Example: Graphing an Ellipse Centered at (h, k) (continued) Graph: ( x1) ( y2) 9 4 1. The center is at ( 1, 2). The vertices are (2, 2) and ( 4, 2). The endpoints of the minor axis are ( 1, 4) and ( 1, 0). Copyright 2014, 2010, 2007 Pearson Education, Inc. 16
Example: An Application Involving an Ellipse A semielliptical archway over a one-way road has a height of 10 feet and a width of 40 feet. We construct a coordinate system with the x-axis on the ground and the origin Will a truck that is 12 feet wide and has a height of 9 feet clear the opening of the archway? at the center of the archway. 6 6 Copyright 2014, 2010, 2007 Pearson Education, Inc. 17
Example: An Application Involving an Ellipse (continued) Using the equation a b equation of the archway as 1, we can express the 1 or 20 10 400 100 1. The edge of the 12-foot-wide truck corresponds to x = 6. We find the height of the archway 6 feet from the center by substituting 6 for x and solving for y. 2 36 y 400 100 1 2 36 y 400 400(1) 400 100 2 36 4y 400 2 4y 364 2 y 91 y 91 9.54 Copyright 2014, 2010, 2007 Pearson Education, Inc. 18
Example: An Application Involving an Ellipse (continued) A semielliptical archway over a one-way road has a height of 10 feet and a width of 40 feet. Will a truck that is 12 feet wide and has a height of 9 feet clear the opening of the archway? We found that the height of the archway is approximately 9.5 feet. The truck will clear the opening of the archway. Copyright 2014, 2010, 2007 Pearson Education, Inc. 19