Download Precalculus – Chapter 8

Precalculus – Chapter 8 - Ellipses
Definition: An ellipse is the set of all points in a plane whose distances from two fixed points have a
constant sum. The fixed points are the foci (F). The line through the foci is the focal axis. The point on
the focal axis midway between the two foci is the center (C). The points on the ellipse that intersect with
the focal axis are the vertices (V).
(Ellipses with Center (0, 0)
Standard Equation
Focal Axis
Foci
Vertices
Semimajor Axis
Semiminor Axis
Pythagorean Relation
x2 y 2

1
a 2 b2
y 2 x2
 1
a 2 b2
Example
Find the vertices and foci of the ellipse 4x2 + 8y2 = 64
Example
Find an equation of the ellipse with foci (0, 3) and (0,3) whose minor axis has length 4.
Sketch the ellipse by hand. An ellipse centered at the origin with its focal axis on a coordinate axis is
symmetric with respect to the origin and both coordinate axes. Such an ellipse can be sketched by first
drawing a rectangle centered at the origin with sides parallel to the coordinate axes and then sketching the
ellipse inside the rectangle. You can do this in two steps: 1. Sketch line segments at x   a (or x  b )
and y  b (or y  a ) and complete the rectangle they determine. 2. Inscribe an ellipse that is tangent
to the inside of the rectangle at ( a, 0) and (0, b) (or ( b, 0) and (0,  a ) ).
Graph the ellipse on your graphing calculator. Just like with the parabola, we must solve for y.