Conic Sections: The Ellipse Dr. Shildneck Fall, 2015

The Ellipse An ellipse is a locus of points such that the sum of the distances between two fixed points (called the foci) is always the same. The axis that runs through the longer part of the ellipse is called the major axis. The points at the ends of the major axis are called the vertices. major axis The axis that runs through the shorter part of the ellipse is called the minor axis. The points at the ends of the minor axis are called the co-vertices.

(just always do bigger minus smaller) Equation of an Ellipse (h, k) = center of the ellipse a = horizontal distance from center to ellipse b = vertical distance from center to ellipse c = distance from center to the foci, where c2 = a2 – b2 or c2 = b2 – a2 (just always do bigger minus smaller)

Example 1: Graph and find the coordinates of the center, vertices, co-vertices and foci. (0, -3) (- 21 , -3) and ( 21 , 3) Co- Vertices Foci Horizontal a2=25 a=5 Vertical b2=4 b=2 Vertices Distance to Foci a2-b2=c2 25-4=c2 21=c2 c = 21 ≈4.6 (-5, -3) and (5, 3)

Writing the Equation of an Ellipse Determine the center (h, k) Determine the values of a and b. Given a graph Given the vertices and co-vertices Given a vertex/co-vertex and focus Plug the values of h, k, a, and b into the equation. * It is always helpful to sketch a quick picture!

Example 2. Write the equation of an ellipse with a vertex at (6, 0) and a co-vertex at (0, 3).

Example 3. Write the equation of an ellipse centered at the origin with a vertex at (5, 0) and focus at (2, 0).

Example 4. Write the equation of an ellipse centered at (3, 5) with a vertex at (9, 5) and co-vertex at (3, 7).

Example 5. Write the equation of an ellipse with a vertex at (-2, 2) and a co-vertex at (1, 4).

ASSIGNMENT Alternate Text (from Dr. Shildneck’s website) Page 677 Vocabulary #1-3 Exercises #1-6, 7, 9, 11, 23, 25, 27, 29, 33, 35, 39, 40, 50, 52