CONICS
Ellipses
Definition

An ellipse is the set of all points in the plane, the sum of whose distances from two fixed points,
called foci, is constant.
The parent graph of an ellipse
is shown at right and is centered
at the origin. An ellipse has two
axes of symmetry. The longer
segment is called the major axis
and the shorter segment is called
the minor axis. The foci are always
found on the major axis. The endpoints of
each axis are the vertices of the ellipse.
Standard Form/Orientation/Description

Orientation


If the major axis is horizontal, then we know that the
larger denominator is under the x equation.
If the major axis is vertical, then we know that the
larger denominator is under the y equation.
Example # 1


Consider the graphed ellipse. Write the equation of the
ellipse in standard form. Find the coordinates of the
foci.
We notice the center (3, 1). We also notice from the
graph that the major axis is vertical so we know the
denominator under the y equation will be larger than
the denominator under the x equation. From the picture,
we see the distance from the center to the circle at (7,1)
is 4 and the vertical distance to (3,-4) is 5
b x  3 g b y  1g
2



16
+
25
2
=1
Now from our chart, we know that the foci will be
2
2
2
(h, k±c). We know a=5 and b= 4, so using c = a - b
So c 2 = 9 which means c = 3
The foci are (3, 4) and (3, -2)
Be careful. This
is not the Pythagorean Theorem
Example # 2

b
g b g
2
For the equation x  3 + y  4
100
2
=1, find the coordinates of the center,
64
foci, and vertices of the ellipse. Then graph the equation.


Center (-3, 4) a=10 b= 8 so c = ±6. Since 100 is larger than 64, this
ellipse has a horizontal major axis and the foci are (h ± c, k) or (3, 4) and (9,4). The major axis vertices will be (-3+10, 4) (-3-10, 4) which are (7,4) &
(-13,4)
The minor axis vertices are (-3, 4+8) (-3, 4-8) which are (-3,12) & (-3, -4)
Example #3

Find the coordinates of the center, the foci, and the
vertices of the ellipse with the equation. Then graph it
2
2
4x +9y -8x-54y+49=0
The only difference between this example and the
previous example is that you must fist get it in standard
form using complete the square.
Work with your partner to find the center, foci and
vertices. Then graph the equation.
Answer
Eccentricity


The eccentricity of an ellipse, denoted by e , is a
measure that describes the shape of an ellipse.
When it is close to zero, the foci are close to the
center. When it is close to 1, the foci are far from
the center of the ellipse. As the value approaches 0,
the ellipse approaches a circle. As the value
approaches 1, the ellipse approaches a line.
e
= c/a
Eccentricity
HW #15
Section 10-3
Pp. 638-641
#10-21 odds, 56, 57
