1. 9.1.1 – Conic Sections; The Ellipse

2. In math, we define a “conic section” given the equation
From the above equation, we have several different types of conics we may define
The first, is known as an ellipse
If AC > 0, then the conic is an ellipse

3. Ellipse There are several properties and features to an ellipse
There exist two points in the plane for which their sum of distances, d1 and d2, to two foci, is a fixed constant

4. Equation of an Ellipse
In terms of Ellipses, we may have one of two types
Major Axis = line segment extending from one end (extreme) of an ellipse to the other and passing through the two foci and center
Length = 2a
Minor Axis = axis perpendicular to major axis
Length = 2b
Centered at Origin;

5. Origin Equations
If an ellipse is centered at the origin, and the major axis is horizontal, then the equation is;
If an ellipse is centered at the origin, and the major axis is vertical, then the equation is;

6. How do I tell if the major axis is vertical or horizontal?
If the coefficient below y is GREATER than the coefficient below x, then the graph is stretched vertically; major axis would be vertical
If the coefficient below x is GREATER than the coefficient below y, then the graph is stretch horizontally; major axis would be horizontal
Major Axis Length = 2a
Minor Axis Length = 2b

7. Foci
To identify the foci, or the points that form a constant, we can use the following formula

8. To graph a standard ellipse, we will do the following
1) Determine major axis (for reference)
2) Find x and y intercepts
3) Plot the 4 “vertices”
4) Solve for foci and plot them

9. Example. Graph the ellipse

10. Example. Graph the ellipse

11. Center NOT at origin
Just like most other cases, similar to circles, not all ellipses will be centered at the origin
The new form is given as;
where (h,k) is the center.

12. When graphing with a different center, it’s best to determine the lengths of the major and minor axis
Just remember, major corresponds to largest coefficient; minor corresponds to smallest coefficient
Length of axis starts from center

13. Example. Graph the ellipse

14. Example. Graph the ellipse

15. Assignment
Pg. 706
13-20 all
21-29 odd