1. 10.3
Ellipses

2. 10.3 Ellipses
Center point
(h,k)
Focus point a b F F c V V a
Minor axis
Major axis
An ellipse is the set of all points (x,y), the sum of whose distances from two distinct points (foci) is constant.
a2 = b2 + c2

3. Standard Equation of an Ellipse
Horz. Major axis
Vert. Major axis
(h,k) is the center point.
The foci lie on the major axis, c units from the center.
c is found by c2 = a2 - b2
Major axis has length 2a and minor axis has length 2b.

4. Sketch and find the Vertices, Foci, and Center point.
x2 + 4y2 + 6x - 8y + 9 = 0
First, write the equation in standard form.
(x2 + 6x ) + 4(y2 - 2y ) = -9
(x2 + 6x + 9) + 4(y2 - 2y + 1) =
(x + 3)2 + 4(y - 1)2 = 4
C (-3,1)
V (-1,1) (-5,1)

5. c2 = a2 - b2
C (-3,1)
c2 = 4 - 1
V (-1,1) (-5,1)
Foci are:

6. Eccentricity e of an ellipse measures the ovalness of the
ellipse. e = c/a
In the last example, what is the eccentricity?
The smaller or closer to 0 that the eccentricity is, the more
the ellipse looks like a circle.
The closer to 1 the eccentricity is, the more elongated it is.

7. Find the center, vertices, and foci of the ellipse given by
4x2 + y2 - 8x + 4y - 8 = 0
First, put this equation in standard form.
4(x2 - 2x + 1) + ( y2 + 4y + 4) =
4(x - 1)2 + (y + 2)2 = 16
C( , )
a =
b =
c =
Vertices ( , ) ( , )
Foci ( , ) ( , )
e =
Sketch it.

8. Assignment: 1-6 all, odd