1. Algebra 2 Warm up
3.10.14
Find the center and radius of the circle and
then graph it:
x  y  8 x  6 y  11
2
2

2. And now it’s time for..
The Lame Joke of the day..
What did Sushi A say to Sushi B?
Wasabi!
(What’s up B, get it?)

3. And now it’s time for..
The Lame Joke of the day..
What did one ocean say to the other
ocean?
Nothing , it just waved

4. THE

5. An ellipse is the set of all points such that the sum of the
distances from two fixed points, called the foci, is a constant.
You can draw an ellipse by taking two push pins in
cardboard with a piece of string attached as shown:
The place where each
pin is is a focus (the
plural of which is
foci). The sum of the
distances from the
ellipse to these points
stays the same
because it is the
length of the string.

6. Minor Axis
The Ellipse
Focus 1
Point
Major Axis
Focus 2
PF1 + PF2 = constant
3.4.2

7. PARTS OF AN ELLIPSE
The major axis is in the direction of the longest part of the
ellipse
major axis
foci
minor axis
center
The vertices are at the ends of the major axis
The foci are always on the major axis
foci

8. The standard equation for an ellipse centered at the origin:
2
2
x
y
2
2
2
 2  1, where a  b  0 and a  b  c
2
a
b
The values of a, b, and c tell us about the size of our ellipse.
b
a
c
b
a
c

9. x2 y2
Find the vertices and foci and graph the ellipse:

1
9
4
From the center the
The ends of
this axis are
the vertices
ends of major axis
are "a" each
direction. "a" is the
square root of
this value
a
a
b
(-3, 0)

We can
now draw
the ellipse
5 ,0

b 5 ,0
(3, 0)
To find the foci, they are
"c" away from the center
in each direction. Find "c"
by the equation:
c  9  4  5 c  5  2.2
2
From the
center the
ends of minor
axis are "b"
each direction.
"b" is the
square root of
this value
c2  a2  b2

10. 2
2 22 2
In
2
x y yy22
An ellipse can have a vertical major axis. x xx  y  1
 1 1
  222  1
22 2
that case the a2 is under the y2
1 4a

1 bb 164
You can tell which value is a because
a2 is always greater than b2
From the
(0, 4)
Find the
center, the
equation of
vertices
the ellipse
are 4 each
shown
way so "a"
is 4.
(0, -4)
From the
center the
ends of minor
axis are 1 each
direction so
"b" is 1
To find the foci, they are
"c" away from the center
in each direction along the
major axis. Find "c" by the
equation:
c2  a2  b2
2
c  16  1  15 c  15  3.9

11. Graph the ellipse:
x2 y2

1
16 64
Is the ellipse horizontal or vertical?
Vertical because the larger number is under “y”
a is always the larger number:
a  64  8
b  16  4

12. Graph the ellipse:
x 2  25 y 2  25
This equation is not in standard form (equal to 1) , so we must
divide both sides by 25.
x2
2
 y 1
25
Is the ellipse horizontal or vertical?
Horizontal because the larger number is under “x”
a is always the larger number.
a  25  5
b  1 1

13. Graph each ellipse:
9 x 2  y 2  36
x 2  25 y 2  25

14. There are many applications of
ellipses.

15. A particularly interesting one is the whispering gallery.
The ceiling is elliptical and a person stands at one
focus of the ellipse and can whisper and be heard by
another person standing at the other focus because all
of the sound waves that reach the ceiling from one
focus are reflected to the other focus.