Ch 4: The Ellipse Objectives Use the standard and general forms of the equation of an ellipse graph ellipses ©2002 Roy L. Gover (www.mrgover.com)

Applications The Ellipse: Planets travel in an elliptical path with the sun at a focus Electrons travel in elliptical pattern about the nucleus of an atom Applications The Ellipse:

Definition An ellipse is the locus (set) of all points such that the sum of the distances from two points called foci is constant.

P1(x1,y1) Vertex Major Axis Minor Axis Focus Center

Important Idea An ellipse has 4 vertices An ellipse has 2 axes-the longer is the major axis; the shorter is the minor axis An ellipse has 2 foci located on the major axis

Definition The standard form of the equation of an ellipse when the major axis is parallel to the x-axis

Definition c b a Center at(h,k) An ellipse with major axis parallel to x-axis

Definition The standard form of the equation of an ellipse when the major axis is parallel to the y-axis

Definition An ellipse with major axis parallel to y-axis a c b Center:at (h,k) c An ellipse with major axis parallel to y-axis

2a is the length of the major axis; 2b is the length of the minor axis Summary The center of an ellipse is at (h,k) 2a is the length of the major axis; 2b is the length of the minor axis c is the distance from the center to each focus

Important Idea a>b a b (h,k) c

Try This Name the parts of an ellipse:

Try This What letter in the standard equation represents the indicated lengths?

Example For the following equation, find the coordinates of the center, foci and vertices, then graph:

Example For the following equation, find the coordinates of the center, foci and vertices, then graph:

Important Idea The direction of the major axis is determined by the larger denominator. The larger denominator is always a2 in the standard equation.

Important Idea If the larger denominator is under the x term, the ellipse is “fat”; if the larger denominator is under the y term, the ellipse is “skinny”

Try This For the following equation, find the coordinates of the center, foci and vertices, then graph:

Solution Center:(0,-4) Foci: ± Vertices: (±6,-4) (0,1),(0,-9)

Example For the following equation, find the coordinates of the center, foci and vertices, then graph:

Try This For the following equation, find the coordinates of the center, foci and vertices, then graph:

Solution Center:(1,-3) Foci:(1,-3 ± ) Vertices:(0,-3),(2,-3) (1,-1),(1,-5)

Lesson Close Describe the meanings of the values a, b, c, h & k. What is wrong? a=5 ; b=6