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The Ellipse Analytic Geometry Section 3.3 1 Definition of “ellipse” • An ellipse is the set of all points in a plane such that the distance from two fixed points (foci) on the plane is a constant. 2 Equation of the Ellipse • The equation of an ellipse with its center at the origin has one of two forms: x2 y 2 2 1 2 a b or • The position of the a2 (under the x or y) tells you whether the horizontal or the vertical axis is the major axis of the ellipse. y2 a2 2 1 2 a b 3 Ellipse x2 y 2 1 64 25 • This ellipse has a horizontal major axis that is 16 units long. a 64, a 8 2 2a 16 4 Ellipse x2 y 2 1 64 25 • The minor axis of this ellipse is 10 units in length. b 25, b 5 2 2b 10 5 Foci • The two foci for this ellipse are the two points lying on the horizontal axis that appear to be a little over 6 units from the origin. The origin is the center of the ellipse. The distance from the center to a focus is “c”. 6 • The segments drawn from the two foci to the point (0,5) on the ellipse are each 8 units in length. Their total length is 16 units. This total length is also the length of the major axis. 7 • Two more segments are added, drawn from the foci to the point (2,4.84) on the ellipse. Their lengths are 9.556 and 6.434. The sum of these lengths is again 16 units. 8 • The two latest segments, drawn to the point (7,-2.42) on the ellipse, are 13.463 units and 2.537 units in length, a sum of 16 units. 9 The Ellipse (0,b) P(x,y) (-a,0) (a,0) (-c,0) (c,0) (0,-b) x2 y 2 2 1 2 a b • The ends of the major axis are at (a,0) and (-a,0). • The ends of the minor axis are at (0,b) and (0,-b). • The foci are at (c,0) and (-c,0). 10 The Ellipse (0,b) • The sum of the distances from point P to the foci is 2a. P(x,y) (-a,0) (a,0) (-c,0) (c,0) (0,-b) x2 y 2 2 1 2 a b 2 2 2 c a b • Also, or b a c 2 2 2 11 • A chord through a focus and perpendicular to the major axis is called a latus rectum. • The endpoints of the two latus recti are found using the equivalence : c a b 2 2 2 providing the endpoints b c, a 2 12 Latus Rectum • When the equation of the ellipse is x2 y2 1 64 25 then c 2 64 25 39 and c 6.24 So the endpoints of the latus recti are: 6.24, 3.125 13 The Ellipse y2 x2 1 169 25 • The major axis is the vertical axis with endpoints (0,13) and (0,-13). The endpoints of the major axis are called the vertices. The minor axis has endpoints of (5,0) and (-5,0). 14 y2 x2 1 169 25 • The foci are found using c 2 a 2 b2 c 2 169 25 144 • so the values of c are 12 and -12. The coordinates of the foci are (0,12) and (0,-12). 15 y2 x2 1 169 25 • The endpoints of the latus recti are: b2 c, a 25 12, 13 16 Problem: Determining an equation • Find the equation of the ellipse with foci at (8,0) and (-8,0) and a vertex at (12,0). • First, place these points on axes. • The F and F’ are the foci. F F' 17 Finding the equation of the ellipse with foci at (8,0) and (-8,0) and a vertex at (12,0). • Since the vertex is on the horizontal axis, the ellipse will be of the form: x2 y 2 2 1 2 a b • The values of a and b need to be determined. 18 Finding the equation of the ellipse with foci at (8,0) and (-8,0) and a vertex at (12,0). • If the foci are at 8 and -8, then c = 8. Since a vertex is at (12,0), that means that a = 12. Relating these values to the standard form for an ellipse whose center is at the origin and whose major axis is horizontal, ax by 1 , and the equivalence c2 a 2 b2 2 2 2 applies. Solve for b2 to get b a c In this case, b2 122 82 144 64 80 2 2 2 2 19 Finding the equation of the ellipse with foci at (8,0) and (-8,0) and a vertex at (12,0). • • • • Since b2 80, b 80 8.94 The value of a is 12, and a2 is 144. The value of b is 80 and b2 is 80. So the equation of the ellipse is: 2 2 x y x y 1 1 or 144 80 a 2 b2 2 • 2 20 Ellipse with center at (h,k) • The ellipses with their centers at the origin are just special cases of the more general ellipse with its center at the point (h,k). This more general ellipse has a standard formula of: ( x h) 2 ( y k ) 2 1 2 2 a b or ( y k ) 2 ( x h) 2 1 2 2 a b 21 Problem: Write the equation in standard form • The general form of the equation is: 4x 8 y 4x 24 y 13 0 2 2 • After writing this in standard form, also find the coordinates of the center, the foci, the ends of the major and minor axes, and the ends of each latus rectum. 22 Write 4x2 8 y2 4x 24 y 13 0 in standard form: • First, group the terms with x’s and the terms with y’s, and move the constant to the other side of the equation. 4x 4x 8 y 24 y 13 2 2 23 Write 4x 8 y 4x 24 y 13 0 2 2 in standard form: • Now factor out the coefficient of each squared term. 4( x x) 8( y 3 y) 13 2 2 • Then complete the square for each variable. 1 2 9 2 1 9 4 x x 8 y 3 y 13 4 8 4 4 4 4 2 2 1 3 4 x 8 y 13 1 18 2 2 24 To finish the problem: • Simplify on the right. 2 2 1 3 4 x 8 y 32 2 2 • Then divide each side by 32. 2 2 1 3 4 x 8 y 32 2 2 32 32 32 2 2 1 3 x y 2 2 1 8 4 25 The Ellipse 2 2 1 3 x y 2 2 1 8 4 1 3 • This ellipse has a center at 2 , 2 . • The major axis is 2 8 in length, and the minor axis is 4 in length, so their endpoints are ( 8 ,0) and (- 8 ,0), (0,2) and (0,-2). • The foci are at (2,0) and (-2,0). 26 To finish 2 2 1 3 x y 2 2 1 8 4 • Since the foci are at (2,0) and (-2,0), the endpoints of the latus recti are at b2 4 2, 2 c, 2, a 8 27 Site and Assignment • There’s a neat website that you might want to look at for more on the ellipse. It’s at: http://mathworld.wolfram.com/Ellipse.html • Your assignment, due Monday, is: 3.3: 2, 3, 15, 16, 17, 22, 25, 44 28