January 13, 2015
9.2 Ellipses
Objective: Write the standard form of the equation of an ellipse and
find the eccentricity of an ellipse.
Ellipseset of all points (x, y) in a plane, the
sum of whose distances from two
distinct fixed points (foci) is constant.
d
(x, y)
d
1
2
d + d = constant
1
2
http://www.brightstorm.com/math/precalculus/conic-sections/the-ellipse
Standard Form
Ellipse with horizontal major axis
center: (h, k)
major axis of length 2a
minor axis of length 2b
vertices: (h ± a, k)
covertices: (h, k ± b)
foci: (h ± c, k)
(x-h)2 (y-k)2
=1
+
a2
b2
Ellipse with vertical major axis
center: (h, k)
major axis of length 2a
minor axis of length 2b
vertices: (h, k ± a)
(y-k)
(x-h)
=1
+
covertices: (h ± b, k)
a
b
foci: (h, k ± c)
2
2
2
2
NOTE: In both cases, a > b > 0 and c = a - b .
a = distance from center to vertices
b = distance from center to covertices
c = distance from center to foci
2
2
2
January 13, 2015
Ex1: Find the center, vertices, and foci of the ellipse given by
9x2 + 4y2 = 36.
x2/4 + y2/9 = 1 ⇒ a
(0, 0), the vertices
Ex2: Find the standard form of the equation of the ellipse centered
at the origin with major axis of length 10 and foci at (±3, 0).
We know that
So the equatio
Ex3: Sketch the graph of the following ellipse:
25x2 + 9y2 - 200x + 36y + 211 = 0
So, the center is at (4, -2), a =
vertices, go up and down 5 un
covertices, go right and left 3
January 13, 2015
Ex4: A passageway in a house is to have straight sides and a
semielliptically-arched top. The straight sides are 5 feet tall, and
the passageway is 7 feet tall at its center and 6 feet wide. Where
are the foci to be located to make the template for the arch?
2a = 6
a = 3. b = 7 - 5 = 2. c = 3 - 2
2
2
2
c = √5 ≈2.236.
The foci should be about 2.236 feet right and left of the semiellipse.
Eccentricity
The eccentricity of an ellipse is given by e = c , where 0 < e < 1.
a
measure of the “ovalness” of the ellipse.
If the eccentricity is close to 0, then the ellipse is close
to being circular.
If the eccentricity is close to 1, then the ellipse is
flatter.
*Examples of orbits and their eccentricities p.645