Transcript Ch. 10.4 Ellipses

Ellipses
Date: ____________
Ellipses
Standard Equation of an Ellipse
Center at (0,0)
y
x2
+
2
a
y2
b2
(–a, 0)
(0, b)
=1
O
(0, –b)
(a, 0)
x
Horizontal Major Axis
Co-Vertices
Vertical Major Axis
Vertices
Vertices
a2
>
b2
Co-Vertices
a2 < b2
For example,
An ellipse is the set of
all points P in a plane
such that the sum of
the distances from P to
two fixed points, F1 and
F2, called the foci, is a
constant.
P
P
P
F1
F2
2a
F1P + F2P = 2a
Horizontal Major Axis:
x2
a2
+
y2
b2
y
=1
(–a, 0)
a 2 > b2
a 2 – b2 = c2
(0, b)
O
F1(–c, 0) (0, –b)
length of major axis: 2a
length of minor axis: 2b
Distance from midpoint and foci: c
(a, 0)
x
F2 (c, 0)
Vertical Major Axis:
x2
+
2
a
y2
b2
=1
b2 > a2
y
F1 (0, c)
(–a, 0)
O
b2 – a2 = c 2
length of major axis: 2b
length of minor axis: 2a
(0, b)
F2(0, –c)
Distance from midpoint and foci: c
(a, 0)
x
(0, –b)
Write an equation of an ellipse in standard
form with the center at the origin and with
the given vertex and co-vertex.
(4,0), (0,3)
Co-Vertices: (0,3)
(0,-3)
Vertices : (4,0)
(-4,0)
So a = 4
So b = 3
a² = 16
x2
16
b² = 9
+
y2
9
=1
Find an equation of an ellipse for the given
height and width with the center at (0,0)
h = 32 ft, w = 16 ft
32 ft
Distance b is from the center is 16
16 ft Distance a is from the center is 8
a=8
a² = 64
b = 16 b² = 256
x2
64
+
y2
256
=1
Find the foci and graph the ellipse.
x2
25
a2 = 25
a = ±5
+
y2
9
=1
y
b2 = 9
b = ±3
25 – 9 = c2
16 = c2
±4 = c
(0, 3)
(–5, 0) (–4, 0)
(5, 0)
(4, 0)
(0,-3)
x
Graph the ellipse. Find the foci.
x2
9
y2
25
+
a2 = 9
a = ±3
=1
y
b2 = 25
b = ±5
b2 – a2 = c 2
25 – 9 = c2
16 = c2
±4 = c
(0, 5)
(0,4)
(–3, 0)
(3, 0)
x
(0,-4)
(0,-5)
Write an equation of an ellipse for
the given foci and co-vertices.
Foci: (±5,0), co-vertices: (0,±8)
Horizontal axis
Since c = 5 and b = 8
– =
a2 – 64 = 25
+ 64 + 64
a2 = 89
a2
b2
c2
c² = 25 and b² = 64
x2
+
2
a
x2
89
+
y2
b2
y2
64
=1
=1
9.4 Ellipses
Translated Ellipses
Standard Equation of an Ellipse
Center at (h,k)
(x – h)2
(y – k)2
+
=1
2
2
a
b
(h–a, k)
y
(h, k+b)
(h+a, k)
(h,k)
(h, k–b)
x
Write an equation of the translation.
Center = (2,-5)
h=2
k = -5
Horizontal major axis of length 12,
minor axis of length 8.
Length of major axis is 2a Length of minor axis is 2b
2b = 8
2a = 12
b=4
a=6
a2 = 36
b2 = 16
2
2
2
2
(x
–
2)
(y
+
5)
(x – h)
(y – k)
+
=1
+
=
1
36
16
a2
b2
Find the foci for the ellipse.
4x2 + 9y2 – 16x +18y – 11 = 0
4x2 – 16x + 9y2 + 18y = 11
4
1 =11+16+9
4(x2 – 4x + ____)
+ 9(y2 + 2y + ___)
4(x – 2)2 + 9(y + 1)2 = 36
36
(x – 2)2 (y + 1)2
+
=1
9
4
(x – 2)2
9
+
(y + 1)2
4
=1
Foci = (2 + 2.2,-1)
Foci = (2 – 2.2,-1)
Center = (2,-1)
a2 = 9
a2 > b2
a2 – b 2 = c2
b2 = 4
Horizontal Axis
9 – 4 = c2
5 = c2
±2.2 ≈ c
Foci = (4.2,-1)
and = (-0.2, -1)