Download Slide 1 - Elgin Local Schools

Pre-Calc
Ellipses
Lesson 6.3
If you take a circle---grab hold of a horizontal diameter
At the ends and just stretch it out symmetrically, then
The result is an ellipse with a horizontal major axis.
Equation will be: x2 + y2 = 1
a2
b2
For a situation like this  we say it has a:
i. horizontal major axis  which will be ‘2a’ units long
the coordinates of the endpoints of major axis will
be (a,0) and (- a,0) (If the center is at the origin)
ii. vertical minor axis  which will be ‘2b’ units long;
the coordinates of the endpoints of minor axis will
be (0,b) and (0, - b) (If the center is at the origin)
Now, If you take that circle---grab hold of a veritcal diameter
at each end and just stretch it out symmetrically, then
the result is an ellipse with a vertical major axis.
Equation will look like this: x2 + y2 = 1
b2
a2
For a situation like this  we say it has a:
i. vertical major axis  which will be ‘2a’ units long;
the coordinates of the endpoints of major axis will
be (0,a) and (0,- a) (If the center is at the origin)
ii. horizontal minor axis  which will be ‘2b’ units
long;
the coordinates of the endpoints of minor axis will
be (b,0) and (- b,0) (If the center is at the origin)
Now I know what you are thinking??
How do you determine which is which? It all depends on
the values located under the x2 and y2  whichever
number is the greater is  a2
Now an ellipse has two other special points that are always
located on the ‘major axis’  These points are called
the ‘foci’ and the variable associated with these points
is ‘c’. So depending on whether your major axis is
horizontal or vertical is how you label the focus point!
(c,0) & (-c,0) or (0,c) & (0,- c)
Also the endpoints on your major axis  either
(a,0) & (-a,0) or (0,a) & (0,- a) are called the:
Vertices of the ellipse!!!!!!
Now to help determine the values of ‘a’, ‘b’, and ‘c’
We use a variation of an old friend:
a2 - b2 = c2
Example 1:
Find an equation of the ellipse with center at the origin,
one vertex at (0,5) and one focus at (0,2).
Sketch the ellipse and label the vertices, and the
endpoints of the minor axis.
Example 2:
Find an equation of an ellipse with center at the origin,
vertex at (4,0) and a minor axis that is 4 units long.
b. Find the coordinates of its ‘foci’.
Example 3:
Sketch the graphs of 4x2 + y2 = 64 and
x + y = 4 on the same set of axes. Determine how
many points of ‘solution’ there are to this system
and ‘solve this system’ ALGEBRAICALLY!!!!!!