Graph and write equations of Ellipses. What you should learn: 10.4 Ellipses Goal 1 Graph and write equations of Ellipses. Goal 2 Identify the Vertices, co-vertices, and Foci of the ellipse. 10.4 Ellipses

An Ellipse is a set of points such that the distance between that point and two fixed points called Foci remains constant P d1 d2 f1 f2 d1 + d2 = constant 10.4 Ellipses

The line that goes through the Foci is the Major Axis. The midpoint of that segment between the foci is the Center of the ellipse (c) The intersection of the major axis and the ellipse itself results in two points, the Vertices (v) The line that passes through the center and is perpendicular to the major axis is called the Minor Axis The intersection of the minor axis and the ellipse results in two points known as co-vertices 10.4 Ellipses

Graphing and Writing Equations of Ellipses An ellipse is the set of all points P such that the sum of the distances between P and two distinct fixed points, called the foci, is a constant. P • • The line through the foci intersects the ellipse at two points, the vertices. d 1 d 2 The line segment joining the vertices is the major axis, and its midpoint is the center of the ellipse. focus focus d 1 + d 2 = constant The line perpendicular to the major axis at the center intersects the ellipse at two points called co-vertices. The line segment that joins these points is the minor axis of the ellipse. The two types of ellipses we will discuss are those with a horizontal major axis and those with a vertical major axis. 10.4 Ellipses

• • • • • • + + Ellipse with horizontal major axis y y vertex: (0, a) vertex: (0, –a) • co-vertex: (0, b) (0, –b) • focus: (0, –c) (0, c) • co-vertex: (b, 0) (–b, 0) vertex: (–a, 0) (a, 0) • • • focus: (–c, 0) (c, 0) x x major axis minor axis minor axis major axis Ellipse with horizontal major axis Ellipse with vertical major axis + = 1 x 2 a 2 y 2 b 2 + = 1 x 2 b 2 y 2 a 2 10.4 Ellipses

cv1 F1 F2 v1 c v2 cv2 10.4 Ellipses

Example of ellipse with vertical major axis 10.4 Ellipses

Example of ellipse with horizontal major axis 10.4 Ellipses

Standard Form for Elliptical Equations Major Axis (length is 2a) Minor Axis (length is 2b) Vertices Co-Vertices Horizontal Vertical (a,0) (-a,0) (0,b) (0,-b) (0,a) (0,-a) (b,0) (-b,0) Note that a is the biggest number!!! 10.4 Ellipses

The foci lie on the major axis at the points: (c,0) (-c,0) for horizontal major axis (0,c) (0,-c) for vertical major axis Where c2 = a2 – b2 10.4 Ellipses

WRITING EQUATIONS Write the equation of an ellipse with center (0,0) that has a vertex at (0,7) & co-vertex at (-3,0) Since the vertex is on the y-axis (0,7) a = 7 The co-vertex is on the x-axis (-3,0) b=3 The ellipse has a vertical major axis & is of the form 10.4 Ellipses

IDENTIFYING PARTS Given the equation 9x2 + 16y2 = 144 Identify: foci, vertices, & co-vertices First put the equation in standard form: 10.4 Ellipses

From this we know the major axis is horizontal & a = 4, b = 3 So the vertices are (4,0) & (-4,0) the co-vertices are (0,3) & (0,-3) To find the foci we use c2 = a2 – b2 c2 = 16 – 9 c = √7 So the foci are at (√7,0) (-√7,0) 10.4 Ellipses

assignment Reflection on the Section How can you tell from the equation of an ellipse whether the major axis is horizontal or vertical? Write the equation in standard form: if the larger denominator is under the x, it is horizontal. Under the y, it is vertical. assignment Page 612 # 19 – 67 odd 10.4 Ellipses