5.3 Ellipses

1

We define an ellipse as the set of points which are a fixed distance from two points (the foci), i.e. that the sum of the two distances from any point on the ellipse to the two foci is the same no matter where you are on the ellipse. An ellipse can be constructed using a piece of string and two thumbtacks. Fix the two ends of the string so the string is not tight (very loose).  Then with a pencil pull the string so that the string is tight and move the string around to form the ellipse.

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●

P

F' F

A line that passes through both foci and intersects the ellipse at two points (the vertices) is known as the major axis. The minor axis is a chord that is perpendicular to the major axis. Their point of intersection is the center.The major axis, containing the foci, is always longer than the minor axis.

vertexvertexcenter

Horizontal Major Axis

focus focus ● ●● ● ●

center

focus

focus

vertex

vertex

Vertical Major Axis

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● ●

●

●

5.3 Ellipses

2

Standard Equation: Ellipse with Major Axis on the x AxisThe standard equation of an ellipse with its center at (0,0) and its major axis on the x axis is

2 2

2 2x y 1a b

where a > b.

The vertices are (a,0) and (a,0), and the length of the major axis is 2a.The endpoints of the minor axis are (0,b) and (0,b), and the length of the minor axis is 2b. 2 2 2The foci are at ( c,0) and (c,0), where c a b .

(-c,0) ● ●

(c,0) ●

(-a,0) (a,0)

(0,b)

(0,-b)

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●

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Next slide

5.3 Ellipses

3

To obtain the standard form, divide both sides by 36 and simplify to obtain a “1” on the RHS.

2 2Find the vertices, the endpoints of the minor axis and the foci of

the ellipse 4x 9y 36,and sketch theEx

eam

llple 1.

ipse. 2 2 4x 9y 36 36 36 36

2 2 x y 1 9 4

2 2

2 2x yCompare to the standard form 1 to determine a and b.a b

2 2a 9 and b 4 then a 3 and b 2

x

y

The vertices (endpoints of the major axis) are (-3,0) and (3,0).The endpoints of the minor axis are (0,2) and (0,-2).

2 2 2fociTo find the , use c a b . 2c 9 4

c 5 2.2

The foci lie on the major axis. Therefore t

foche coordinates

of the are 5,0 and i 5,0 .Next Slide

5.3 Ellipses

4

Your Turn Problem #1

x

y vertices: (-5,0), (5,0)endpoints: (0,-2), (0,2)

Answer

foci: 21,0 and 21,0

2 2Find the vertices, the endpoints of the minor axis and the foci ofthe ellipse 4x 25y 100,and sketch the ellipse.

5.3 Ellipses

5

Standard Equation: Ellipse with Major Axis on the y Axis

The standard equation of an ellipse with its center at (0,0) and its major axis on the y axis is

2 2

2 2x y 1a b

where b > a.

The vertices are (0,-b) and (0,b), and the length of the major axis is 2b.The endpoints of the minor axis are (-a,0) and (a,0), and the length of the minor axis is 2a.

2 2 2The foci are at (0,-c) and (0,c), where c b a .

(0,b)

(0,-c) ●

(0, c) ●(-a,0) (a,0)

(0,-b) ●

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●

●

Next slide

5.3 Ellipses

6

To obtain the standard form, divide both sides by 225 and simplify to obtain a “1” on the RHS.

2 2Find the vertices, the endpoints of the minor axis and the foci of

the ellipse 25x 9y 225,and sketch theExample 2.

ellipse. 2 2 25x 9y 225 225

225 225

2 2 x y 1 9 25

2 2

2 2x yCompare to the standard form 1 to determine a and b.a b

2 2a 9 and b 25 then a 3 and b 5

The vertices (endpoints of the major axis) are (0,-5) and (0,5).

The endpoints of the minor axis are (-3,0) and (3,0).

2 2 2fociTo find the , use c b a . 2c 25 9 c 16 4

The foci lie on the major axis. Therefore t

foche coordinates

of the are 0, 4 and i 0,4 .Next Slide

x

y

5.3 Ellipses

7

Your Turn Problem #2

x

y

vertices: (0,-3), (0,3)endpoints: (-1,0), (1,0)

Answer

foci: 0, 2 2 and 0,2 2

2 2Find the vertices, the endpoints of the minor axis and the foci ofthe ellipse 9x y 9, and sketch the ellipse.

5.3 Ellipses

8

Ellipses whose center is not at the origin.The standard form for an ellipse where the center is not at the origin is

2 2

2 2 x h y k 1 a b where the center is (h,k).

If a>b, then the ellipse has a horizontal major axis. If a<b, then the ellipse has a vertical major axis. Also, the foci which lie on the major axis will be a distance of ‘c’ units from the center.

y

x

(h,k)

(h,k+b)

(h,k-b)

● ●

●

●

●(h-a,k)

●(h+a,k)

(h,k+c)

(h,k-c)

●x(h-a,k) (h-c,k)

● (h,k) ● ●(h+c,k) ●

(h+a,k) ●

(h,k+b)

(h,k-b)●

●

y

Next Slide

2 2 2

2 2 2

Note: To find the foci which lie on the major axis: c a b if a b c b a if a b

5.3 Ellipses

9

2 2Find the center, vertices, the endpoints of the minor axis, the foci of

x 2 y 3 the ellipse 1, and sketch

Example

the ellipse.4 1

3.

6Compare to the standard form to determine the center, a and b.

2 2a 4 and b 16 then a 2 and b 4

x

y

The vertices (endpoints of the major axis) are (2,1) and (2,-7).

The endpoints of the minor axis are (0,-3) and (4,-3).

2 2 2fociTo find the , use c b a . 2c 16 4

c 12 2 3 3.5

The center (2,is 3).

(2,-3)

(2,-7)

(2,1)

323,2

323,2

The foci lie on the major axis. Therefore the coordinatesof the are 2, 3 2 3 and 2, 3 2 3 or (2,0.5), (2,

foci6.5)

Next Slide

(0,-3)

(4,-3)

5.3 Ellipses

10

(-1,-1)

(-1,2)(-6,2)

x

● ● ●

●

●y

center: (-1,2) vertices: (-6,2), (4,2)endpoints: (-1,5), (-1,-1)foci: (-5,2), (3,2)

Answer:

(4,2)

(-1,5)

●

(3,2)

●

(-5,2)

Your Turn Problem #3

2 2Find the center, vertices, the endpoints of the minor axis, the foci of

x 1 y 2the ellipse 1, and sketch the ellipse.25 9

5.3 Ellipses

11

Procedure: Writing an ellipse in standard form given the general form.1. Move the constant to the right hand side and rearrange the terms as

follows: (ax2 + cx + __) + (by2 + dy + ___) = -e.

The ellipse in the previous example was given in standard form:

2. Factor out the a from the first trinomial and the b from the second trinomial. Then create two perfect square trinomials using the technique of completing the square to obtain a(x – h)2 + b(y – k)2 = #.

2 2

2 2 x h y k 1 a b

If the ellipse is given in general form, ,0edycxbyax 22

convert it to standard form before graphing. we will need to

3. Divide by the number on the RHS to obtain the standard form,

2 2

2 2 x h y k 1 a b

Next Slide

5.3 Ellipses

12

Example 4. Write the given equation of the ellipse in standard form:Group the x terms separately from the y

terms and move the constant to the RHS. 2 24x 9y 8x 54y 49 0

2 24 x 2x 9 y 6y 49 2 24x 8x 9y 54y 49

Complete the square for both trinomials. The numbers added in the parenthesis are 1 and 9. We need to add the same “value” to the RHS. The value is 4 and 81.

4 811 9

2 24 x 1 9 y 3 36

Write each perfect square trinomial as a binomial squared and add the constants on the RHS.

Factor out the ‘4” from the first grouping and the ‘9’ from the second group. Leave a space at the end of each set of parentheses to add the appropriate number when completing the square.

Finally, divide by 36 on both sides and simplify to obtain the ellipse in standard form.

36 36 36

2 2x 1 y 3Answer : 19 4

Your Turn Problem #4Write the given equation of the ellipse in standard form:

2 25x 4y 50x 24y 141 0 2 2 x 5 y 3Answer: 14 5

5.3 Ellipses

13

2 2Find the center, vertices, the endpoints of the minor axis, the fociExample of

the ellipse 2x 9y 8x 18y 1 0, and sketch the ell 5.

ipse.

a 3 and b 2 1.4

x

y

The vertices (endpoints of the major axis) are (-1,1) and (5,1).

2 2 2fociTo find the , use c b a . 2c 16 4

c 7 2.6

The are 2 7, 1 and 2 7, 1 or (4.6,1), ( 0.

foci6,1)

The center (2is ,1).

(2,1)

1st write in standard form using completing the square. 1y18y9x8x2 22

1 y2y9 x4x2 22 +8+9+1+4 181y92x2 2

121y

92x 22

Now we can find the center, vertices and endpoints.

(5,1)(-1,1)

21,2

21,2

1,72 1,72

Next Slide

endpoints of the minor axisThe are 2,1 2 .

5.3 Ellipses

14

(-3,-2)

(-2,1)(-4,1)

x

●

●

●

● ●

ycenter: (-3,1) vertices: (-3,4), (-3,-2)endpoints: (-4,1), (-2,1)

Answer:(-3,4)

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●

(-3,1) 221 3,-,221 3,- :foci

221 3,-

221 3,-

The EndB.R.1-28-07

Your Turn Problem #5

2 2Find the center, vertices, the endpoints of the minor axis, the foci ofthe ellipse 9x y 54x 2y 73 0, and sketch the ellipse.