An ellipse is an oval-shaped curve that has the appearance of an
elongated circle.
Ellipse
An ellipse is the set of points in a plane for which the sum of
the distances from two fixed points is a given constant. The
two fixed points are the focal points of the ellipse; the line
passing through the focal points is called the axis. The points
of intersection of the axes and the ellipse are called the
vertices.
•
vertex
F
• focal
•
point
y
P
• P(x,y)
F
•
focal
point
• vertex
axis
vertex
F (-c,0)
• focal
•
point
F(c,0)
vertex x
• •
focal
point
There are two standard position ellipses:
An ellipse with axis along the x-axis of the xy plane with
origin midway between the focal points with coordinates
F (c, 0) and F ' (−c, 0)
and vertices V (a, 0) and V ' (−a, 0)
An ellipse with axis along the y-axis of the xy plane with
origin midway between the focal points with coordinates
F (0, c) and F ' (0, − c)
and vertices V (0, a ) and V ' (0, − a)
The distance between the two focal points is 2c
The distance between the two vertices on the major axis is 2a where
a>c>0
The distance between the two vertices on the minor axis is 2b
y
This ellipse is in standard position
with major axis along the x-axis.
The major axis of an ellipse intersects
the foci and the minor axis does not.
The minor axis is the short axis.
(0,b)
•
b
(-a, 0)
•
•(-c, 0)
a
c
•
•(c, 0)• (a, 0)
x
An ellipse in standard position with
major axis along the x-axis has
equation:
(0, -b)
x2 y 2
+ 2 =1
2
a b
y
(0, a)
•
(0, c)
•
c
•
(-b,0)
a
b
(0, -c)
•
(0, -a)•
•
x
(b, 0)
This ellipse is in standard position
with major axis along the y-axis.
The major axis of an ellipse intersects
the foci and the minor axis does not.
The minor axis is the short axis.
An ellipse in standard position with
major axis along the y-axis has
equation:
y 2 x2
+ 2 =1
2
a b
Standard Position Ellipses
An ellipse with focal points (c, 0) and (-c, 0) and vertices (a, 0)
and (-a, 0), where a > c > 0, is in standard position with axis
along the x-axis and has equation
x2 y2
+ 2 =1
2
a
b
where
b=
a2 − c2
Similarly, an ellipse with focal points (0, c) and (0, -c) and
vertices (0, a) and (0, -a), where a > c > 0, is in standard
position with axis along the y-axis and has equation
2
2
y
x
+ 2 =1
2
a
b
where
b=
a2 − c2
2
2
Sketch the graph of the ellipse with equation 9 x + 16 y = 144
and find its focal points.
We must get this ellipse in standard form, so we must
divide both sides by 144 so that the right side of the equation
is 1.
x2 y 2
+
=1
16 9
2
Since the coefficient in the denominator of x is larger
than the coefficient in the denominator of y 2, the
ellipse is in standard position with axis along the x-axis
a 2 = 16 and b 2 = 9
Axis intercepts: (4, 0), (-4, 0) and
(0, 3), (0, -3)
a = 4 and b = 3
Now, use the fact that b = a 2 − c 2 to find the value of c.
b = a2 − c2
b2 = a 2 − c2
Focal Points:
(− 7 ,0) and ( 7 ,0)
b 2 − a 2 = −c 2
a 2 − b2 = c2
a2 − b2 = c
4 2 − 32 = c
16 − 9 = c
± 7=c
Graphing: Since there are no constants being
added or subtracted from x or y there are no
vertical or horizontal shifts. So the ellipse is in
standard position with center at the origin.
2
2
Sketch the graph of the ellipse with equation 25 x + 4 y = 100
and find its focal points.
Find an equation of the ellipse in standard position that has a
vertex at (5,0) and a focal point at (3,0).
Since they say it is in standard position the center is at the origin.
they tell you that the vertex is at (5,0), so this tells you that the
major axis is aligned with the x-axis.
The other vertex is at (-5,0) and the other focal point is (-3,0)
This tells us that a = 5, and c = 3. Using the formula b = a 2 − c 2
we find that b = 4
2
2
The formula that must be used is when a is under x
x2 y 2
+ 2 =1
2
a b
x2 y2
+
=1
25 16
Sketch together
By substitution we get…
or
16 x 2 + 25 y 2 = 400
Find the equation of the ellipse in standard position that has vertex
at (0, 6) and focal point at (0, 2). Sketch the graph.
Sketch the graph of the ellipse and find its focal points and vertices.
9 x 2 − 72 x + 4 y 2 + 16 y + 124 = 0
Complete the square on both terms x and y.
9( x 2 − 8 x) + 4( y 2 + 4 y ) = −124
9( x 2 − 8 x + 16) − 9(16) + 4( y 2 + 4 y + 4) − 4(4) = −124
9( x − 4) 2 + 4( y + 2) 2 = 36
( x − 4) 2 ( y + 2) 2
+
=1
4
9
The dome of the Mormon Tabernacle in Salt Lake City is 250
feet long, 150 feet wide, and 80 feet high, and its longitudinal
cross section is in the shape of an ellipse. The conductor for
performances stands at one end of the focal points of the ellipse,
and recording equipment can be placed at the other. In this way
sound heard by the conductor corresponds very closely to the
sound being recorded. Determine the location of these points.
this is the situation when an xy-plane
is superimposed on a cross section of
the dome. The width of the dome plays
no part in the calculations.
(0,80)
(-125,0)
(125,0)
since the ellipse is positioned this way
we will use the equation for this
standard position: x 2 y 2
a2
+
b2
=1
We will use the formula b = a 2 − c 2 and substitute in our
values for a and b to find c.
80 = 1252 − c 2
c ≈ 96 feet
So, the conductor should stand 125 – 96 = 29 feet from one end of
the building, and the recording equipment should be placed an
equal distance from the other.
The eccentricity of an ellipse tells how the ellipse differs from
a circle.
c
a 2 − b2
eccentricity = =
a
a
for standard position ellipses
Note: as the focal point approaches the origin, the ellipse approaches a circle,
and the eccentricity approaches 0. The ellipse becomes increasingly elongated
as the focal points approach the vertices which occurs when the eccentricity
approaches 1.