Conic Sections
Ellipse
Part 3
1
Additional Ellipse Elements
• Recall that the parabola had a directrix
• The ellipse has two directrices


They are related to the eccentricity
a a2

Distance from center to directrix =
e c
2
Directrices of An Ellipse
• An ellipse is the locus of points such that



The ratio of the distance to the nearer focus to …
The distance to the nearer directrix …
Equals a constant that
is less than one.
• This constant
is the eccentricity.
c
e
a
3
Directrices of An Ellipse
• Find the directrices of the ellipse defined by
x2 y 2

1
49 35
4
Additional Ellipse Elements
• The latus rectum is the distance across the
ellipse at the focal point.


There is one at each focus.
They are shown in red
5
Latus Rectum
• Consider the length of the latus rectum
• Use the equation for
an ellipse and
solve for the y value
when x = c

Then double that
distance
Length =
2b 2
a
6
Try It Out
• Given the ellipse
 x  3
16
2
y  2


9
2
1
• What is the length of the latus rectum?
• What are the lines that are the directrices?
7
Graphing An Ellipse On the TI
• Given equation of an ellipse


 x  3   y  2   1
25
36
2
2
We note that it is not a
function
Must be graphed in two portions
• Solve for y
8
Graphing An Ellipse On the TI
• Use both results
Set resolution to 1 to
close gaps between
upper and lower portion
9
Area of an Ellipse
• What might be the area of an ellipse?
• If the area of a circle is
 r
2
…how might that relate to the area of the
ellipse?

An ellipse is just a unit circle that has been
stretched by a factor A in the x-direction, and a
factor B in the y-direction
10
Area of an Ellipse
• Thus we could conclude that the are of an
ellipse is
a  b 
• Try it with
2
2
x
y

1
36 25
• Check with a definite integral (use your
calculator … it’s messy)
11
Assignment
•
•
•
•
Ellipses C
Exercises from handout 6.2
Exercises 69 – 74, 77 – 79
Also find areas of ellipse
described in 73 and 79
12