Ellipses
Date: ____________
9.4 Ellipses
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
Vertices
Vertical Major Axis
Vertices
Co-Vertices
Graph the ellipse. Find the vertices
and
co-vertices.
2
2
x
25
+
a2 = 25
a = ±5
y
9
=1
y
b2 = 9
b = ±3
(0, 3)
(–5, 0)
(5, 0)
x
Horizontal Major Axis
Vertices: (–5, 0) and (5,0)
Co-vertices:(0, 3) and (0,-3)
(0,-3)
Graph the ellipse. Find the vertices
and co-vertices.
x2
9
+
a2 = 9
a = ±3
y2
25
=1
y
b2 = 25
b = ±5
(0, 5)
(–3, 0)
(3, 0)
x
Vertical Major Axis
Vertices: (0,5) and (0,-5)
Co-vertices: (-3,0) and (3,0)
(0,-5)
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, 0)
y
(0, k+b)
(h+a, 0)
(h,k)
(0, k–b)
x
Graph the ellipse
(x – 2)2
(y + 5)2
+
=1
36
16
y
Center = (2,-5)
a2 = 36
a = ±6
b2 = 16
b = ±4
Horizontal Major Axis
Vertices: (8,-5) and (-4,-5)
Co-vertices: (2,-1) and (2,-9)
x
Graph the ellipse
(x + 3)2
(y + 1)2
+
=1
25
81
y
Center = (-3,-1)
a2 = 25
a = ±5
b2 = 81
b = ±9
Vertical Major Axis
Vertices: (-3,8) and (-3,-10)
Co-vertices: (-8,-1) and (2,-1)
x
Write the equation of the ellipse in standard form.
Graph. Find the center, vertices, and co-vertices.
4x2 + 25y2 = 100
y
100
x2
25
+
y2
4
=1
Center = (0,0)
a2 = 25
a = ±5
b2 = 4
b = ±2
Vertices: (-5,0) and (5,0)
Co-vertices: (0,2) and (0,-2)
x
Write the equation of the ellipse in standard form.
Graph. Find the center, vertices, and co-vertices.
25x2 + 3y2 = 75
y
75
x2
y2
+
=1
3
25
Center = (0,0)
a2 = 3
a ≈ ±1.73
b2 = 25
b = ±5
Vertices: (0,5) and (0,-5)
Co-vertices: (-1.73,0) and
(1.73,0)
x
Write the equation of the ellipse in standard form.
Graph. Find the center, vertices, and co-vertices.
x2 + 9y2 – 4x + 54y + 49 = 0
x2 – 4x + 9y2 + 54y = -49
9 = -49 +4 +81
4 + 9(y2 + 6y + ___)
x2 – 4x + ____
(x – 2)2 + 9(y + 3)2 = 36
36
(x – 2)2
(y + 3)2
+
=1
36
4
(x – 2)2
(y + 3)2
+
=1
36
4
y
Center = (2,-3)
a2 = 36
a = ±6
b2 = 4
b = ±2
Vertices: (-4,-3) and (8,-3)
Co-vertices: (2,-1) and (2,-5)
x
Write the equation of the ellipse in standard form.
Graph. Find the center, vertices, and co-vertices.
4x2 + y2 +24x – 4y + 36 = 0
4x2 + 24x + y2 – 4y = -36
9
4 = -36 +36 + 4
4(x2 + 6x + ____)
+ y2 – 4y + ___
4(x + 3)2 + (y – 2)2 = 4
4
(x + 3)2 (y – 2)2
+
=1
1
4
(x + 3)2
1
+
(y – 2)2
4
=1
y
Center = (-3,2)
a2 = 1
a = ±1
b2 = 4
b = ±2
Vertices: (-3,4) and (-3,0)
Co-vertices: (-4,2) and (-2,2)
x
Write the equation of the ellipse in standard form.
Graph. Find the center, vertices, and co-vertices.
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
y
Center = (2,-1)
a2 = 9
a = ±3
b2 = 4
b = ±2
Vertices: (5,-1) and (-1,-1)
Co-vertices: (2,1) and (2,-3)
x
9.4 Ellipses
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
9.4 Ellipses
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
(a, 0)
x
F2 (c, 0)
9.4 Ellipses
Vertical Major Axis:
x2
+
2
a
y2
b2
=1
b 2 > a2
y
F1 (0, c)
(–a, 0)
O
b 2 – a2 = c2
length of major axis: 2b
length of minor axis: 2a
(0, b)
F2(0, –c)
(a, 0)
x
(0, –b)
Find the foci.
x2
25
+
y2
9
=1
y
25 – 9 = c2
16 = c2
±4 = c
(–4, 0)
(4, 0)
x
Find the foci.
x2
9
+
y2
25
25 – 9 = c2
16 = c2
±4 = c
=1
y
(0,4)
x
(0,-4)
Find the foci.
x2
100
+
y2
36
=1
y
100 – 36 = c2
64 = c2
±8 = c
(–8, 0)
(8, 0)
x
Find the foci.
(x – 4)2
(y – 3)2
+
=1
16
25
25 – 16 = c2
9 = c2
±3 = c
y
(4, 6)
(4, 0)
x
Find the foci.
(x + 1)2
(y + 2)2
+
=1
4
16
16 – 4 = c2
12 = c2
±3.5 ≈ c
y
(-1,1.5)
x
(-1,-5.5)