Section 9.1 The Ellipse

Overview Conic sections are curves that result from the intersection of a right circular cone—think ice cream cone—and a plane—think sheet of paper. Two of the sections, the circle and the parabola, have been discussed previously. We will re-introduce the parabola later on in the chapter.

The Ellipse An ellipse is the set of all points in a plane the sum of whose distances from two fixed points is a constant. The two fixed points are called foci (plural of focus). The midpoint of the line segment containing the foci is the center of the ellipse.

A Picture

Parts of the ellipse The line that passes through the foci intercepts the ellipse at two points, called vertices (plural of vertex). The line segment with vertices for endpoints is called the major axis. The line segment with endpoints on the ellipse, through the center and perpendicular to the major axis is called the minor axis.

Another Picture

Nomenclature The distance from the center of the ellipse to either of the vertices is a (it follows that the length of the major axis is 2a). The distance from the center to either endpoint of the minor axis is b (it follows that the length of the minor axis is 2b). The distance from the center to a focus is c.

Two equations The standard form of the equation of an ellipse depends on whether the major axis is horizontal Or vertical

One More Picture

Important information b 2 = a 2 – c 2 (equivalently, c 2 = a 2 – b 2 ) When all else fails, draw a picture!

Examples Graph the ellipse and locate the foci:

More Examples—Draw The Picture! Find the standard form of the equation of the ellipse satisfying the following conditions: 1.Foci: (-5, 0) and (5, 0); vertices: (-8, 0) and (8, 0) 2.Major axis vertical with length 12; length of minor axis = 10; center: (-7, -1).

One More…. Convert the equation to standard form by completing the square on x and y. Then graph the ellipse and give the location of its foci.