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1.3 Ellipse
MATH 36
1
directrix
focus
P1
P2
F
Q1
Q2
Given the eccentricity e of a conic section, the conic is
parabola if e = 1;
ellipse if 0 < e < 1;
hyperbola if e > 1.
PQ
principal axis
vertex
Non-degenerate Conic
2
Objectives: At the end of this section students should be able to:
• give the standard equation of an ellipse;
• identify parts of an ellipse;
• sketch the graph of an ellipse.
Ellipse
3
An ellipse is the set of all points on the plane, the sum of whose distances from two fixed points is a constant.
The fixed points referred to are called the foci of the ellipse.
Ellipse
4
EllipseSuppose the foci have coordinates F1(c,0) and F2(-c,0), 2a is the constant sum and if P(x,y) is any point in the ellipse then
a2 PF PF ___
2
___
1 =+
F1(c,0)F2(-c,0)
P(x,y)
5
Ellipsea2 PF PF
___
2
___
1 =+
F1(c,0)F2(-c,0)
P(x,y)
( ) ( ) a2ycxycx 2222 =++++−⇒
( ) ( ) 2222 ycxa2ycx ++−=+−⇒
( ) ( ) ( ) 2222222 ycxycxa4a4ycx +++++−=+−⇒6
Ellipse
F1(c,0)F2(-c,0)
P(x,y)
( )( )222
22222
caa
yaxca
−=
+−⇒
(READING ASSIGNMENT!!!)
In the triangle ,FPF 21∆
FF PF PF _____
21
___
2
___
1 >+
7
Ellipse
F1(c,0)F2(-c,0)
P(x,y)c2a2 >⇒
ca >⇒0ca 22 >−⇒
.bca 222 =−Let
.bayaxb 222222 =+Then
1b
y
a
x2
2
2
2=+
8
More Parts….1
b
y
a
x2
2
2
2=+Consider
(a,0)(-a,0)
(0,b)
(0,-b)
The x-intercepts of the graph are a and –a.
The y-intercepts of the graph are b and –b.
The principal axis is the x-axis.
The points (a,0) and (-a,0) are the vertices of the ellipse.
9
More Parts…
(a,0)(-a,0)
(0,b)
(0,-b)
The line segment joining the vertices is called the major axis of the ellipse.
The line segment joining the points (0,b) and (0,-b) is called the minor axis of the ellipse.
The intersection of the major axis and the minor axis of the ellipse is called its center. 10
Some Remarks… Since a2 – c2 = b2 , then a > b. Hence, the
major axis of the ellipse is always longer than its minor axis.
Since a > c, then (c/a) < 1. This ratio is the eccentricity of the ellipse while the directrices of the ellipse are at the lines
.c
a
e
ax
2±=±=
11
center: (0, 0)
principal axis: x-axis
vertices: (a,0) and (-a,0)
foci: (c,0) and (-c,0) with c2 = a2 - b2
The equation 1b
y
a
x2
2
2
2=+
is the standard equation of the ellipse with
, where ,ba >
endpoints of the minor axis: (0,b) and (0,-b)
equations of directrices: c
a
e
ax
2±=±=
(a,0)(-a,0)
(0,b)
(0,-b)
Standard Equation
12
center: (0, 0)
principal axis: y-axis
vertices: (0,a) and (0, -a)
foci: (0, c) and (0, -c) with c2 = a2 - b2
, whereThe equation 1b
x
a
y2
2
2
2=+
is the standard equation of the ellipse with
,ba >
endpoints of the minor axis: (b, 0) and (-b, 0)
equations of directrices: c
a
e
ay
2±=±=
Standard Equation
(0,-a)
(0,a)
(b,0)(-b,0)
13
14
Example 1. Given the ellipse with equation
determine the principal axis, vertices, endpoints of the minor axis, lengths of the major and minor axes, foci, eccentricity and equations of directrices. Draw also a sketch of the ellipse.
14
y
9
x 22
=+
15
SOLUTION
center: (0,0)
principal axis: x-axis
vertices:(3,0) and (-3,0)
endpoints of minor axis:
(0,2) and (0,-2)
3a =
14
y
9
x 22
=+
22 bac −=2b =
( )0,3( )0,3−
( )2,0
( )2,0 −
549c =−=
16
SOLUTION
14
y
9
x 22
=+
foci:
eccentricity:
equation of the directrices:
( )0,5±
3
5
5
9x ±=
( )0,3− ( )0,5( )0,5−
( )2,0
( )2,0 −
( )0,3
17
Example 2. Given the ellipse with equation
determine the principal axis, vertices, endpoints of the minor axis, lengths of the major and minor axes, foci, eccentricity and equations of directrices. Draw also a sketch of the ellipse.
116
y
4
x 22
=+
18
SOLUTION
center: (0,0)
principal axis: y-axis
vertices:(0,4) and (0,-4)
endpoints of minor axis:
(2,0) and (-2,0)
4a =
116
y
4
x 22
=+
2b =
( )0,2( )0,2−
( )4,0
( )4,0 −
12416c =−=32=
19
SOLUTION
116
y
4
x 22
=+
foci:
eccentricity:
equation of the directrices:
( )32,0 ±
2
3
a
c =( )0,2( )0,2−
( )4,0
( )4,0 −3
8
32
16
c
ay
2
±=±=±=
( )32,0
( )32,0
20
SOLUTION
is equivalent to
.125
y
4
x 22
=+
Example 3. Determine the standard equation of the given ellipse
and give the properties as done in the previous examples.
100425 22 =+ yx
100y4x25 22 =+
21
22
Let be the center of the ellipse
2a be the distance between the vertices
2b be the length of its minor axis
2c be the distance between its foci.
( )k,h
h
k
h
k
23
( ) ( )1
b
ky
a
hx2
2
2
2=−+−
( ) ( )1
b
hx
a
ky2
2
2
2=−+−
Standard equation of the ellipse with center at (h, k) is given by
if the principal axis is a horizontal line
if the principal axis is a vertical line.
24
Example 4. Given the standard equation of the
Determine the (a) principal axis, (b) vertices, (c) endpoints of the minor axis, (d) lengths of the major and minor axes, (e) foci, (f) eccentricity, and (g) equations of directrices. Draw also a sketch of the ellipse.
( ) ( )1
100
3y
36
2x 22=+++
25
SOLUTION
center: (-2, -3)
principal axis: y = -3
vertices:(-2,7) and (-2,-13)
endpoints of minor axis:
(-8,-3) and (4,-3)
10a =
( ) ( )1
100
3y
36
2x 22
=+++
6b =
( )3,4 −( )3,8 −−
( )13,2 −−
( )7,2−
6436100c =−=8= ( )3,2 −−
26
SOLUTION
foci:
eccentricity:
equation of the directrices:
( ) ( )13,2,5,2 −−−
( ) ( )1
100
3y
36
2x 22
=+++
( )3,4 −( )3,8 −−
( )13,2 −−
( )7,2−
( )3,2 −−
( )5,2−
( )11,2 −−
5
4
10
8
a
c ==
8
1003
c
aky
2
±−=±=
27
Example 5. Given the standard equation of the
Determine the (a) principal axis, (b) vertices, (c) endpoints of the minor axis, (d) lengths of the major and minor axes, (e) foci, (f) eccentricity, and (g) equations of directrices. Draw also a sketch of the ellipse.
( ) ( ) 134
4 22
=++−y
x
28
2a =1b =
3c =
21 43 65
-1
-2
-3
-4
-5
-6
SOLUTION
( ) ( ) 13y4
4x 22
=++−
29
Example 6. Determine the standard equation of the ellipse with center at (1,-1), principal axis parallel to the x-axis and the lengths of the major and minor axes are 8 and 4, respectively.
30
Example 7. Write the equation of the ellipse
in standard form.
11189164 22 =−+− yyxx
31
SUMMARY
Center C(h,k)
Vertices V(h±a,k)
Foci F(h±c,k)
Endpoints of minor axis B(h,k±b)
Directrices x = h±a/e
( ) ( )1
b
ky
a
hx2
2
2
2=−+−
32
Center C(h,k)
Vertices V(h,k±a)
Foci F(h,k±c)
Endpoints of the minor axis B(h±b,k)
Directrices y = k±a/e
( ) ( )1
b
hx
a
ky2
2
2
2=−+−
33END
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