2.6ellipses x

Conic Sections

Conic SectionsWe continue with the graphs of Ax2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers (A or B ≠ 0).

Conic SectionsWe continue with the graphs of Ax2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers (A or B ≠ 0). Their graphs are the conic sections as shown below.


Circles and ellipses are enclosed.




If the equation Ax2 + By2 + Cx + Dy = E has A = B so it’s of the form Ax2 + Ay2 + Cx + Dy = E,



If the equation Ax2 + By2 + Cx + Dy = E has A = B so it’s of the form Ax2 + Ay2 + Cx + Dy = E, dividing by A, we obtain 1x2 + 1y2 + #x + #y = #,



If the equation Ax2 + By2 + Cx + Dy = E has A = B so it’s of the form Ax2 + Ay2 + Cx + Dy = E, dividing by A, we obtain 1x2 + 1y2 + #x + #y = #,and its graph is a circle.



If the equation Ax2 + By2 + Cx + Dy = E has A = B so it’s of the form Ax2 + Ay2 + Cx + Dy = E, dividing by A, we obtain 1x2 + 1y2 + #x + #y = #,and its graph is a circle.


Circles: 1x2 + 1y2 + #x + #y = #

Conic SectionsIf an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,but A and B having the same sign,


Circles: 1x2 + 1y2 + #x + #y = #

Conic SectionsIf an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,but A and B having the same sign, after dividing by A, we obtain 1x2 + ry2 + #x + #y = #, with r > 0.


Circles: 1x2 + 1y2 + #x + #y = #

Conic SectionsIf an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,but A and B having the same sign, after dividing by A, we obtain 1x2 + ry2 + #x + #y = #, with r > 0. This is an ellipse.


Circles: 1x2 + 1y2 + #x + #y = #

Conic SectionsIf an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,but A and B having the same sign, after dividing by A, we obtain 1x2 + ry2 + #x + #y = #, with r > 0. This is an ellipse.


Circles: 1x2 + 1y2 + #x + #y = #

Ellipses: 1x2 + ry2 + #x + #y = # (r > 0)

Conic SectionsIf an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,but A and B having the same sign, after dividing by A, we obtain 1x2 + ry2 + #x + #y = #, with r > 0. This is an ellipse.Geometrically, the ellipses are “squashed” circles and the r controls the compression or extension factor along the vertical or the y-direction of the circles.


Circles: 1x2 + 1y2 + #x + #y = #

Ellipses: 1x2 + ry2 + #x + #y = # (r > 0)

Conic SectionsIf an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,but A and B having the same sign, after dividing by A, we obtain 1x2 + ry2 + #x + #y = #, with r > 0. This is an ellipse.Geometrically, the ellipses are “squashed” circles and the r controls the compression or extension factor along the vertical or the y-direction of circles.


Circles: 1x2 + 1y2 + #x + #y = #

Ellipses: 1x2 + ry2 + #x + #y = #

Ellipses also are horizontally stretched or compressed circles.

(r > 0)

Conic SectionsIf an equation Ax2 + By2 + Cx + Dy = E has A ≠ B,but A and B having the same sign, after dividing by A, we obtain 1x2 + ry2 + #x + #y = #, with r > 0. This is an ellipse.Geometrically, the ellipses are “squashed” circles and the r controls the compression or extension factor along the vertical or the y-direction of circles. Let's look at ellipses.


Circles: 1x2 + 1y2 + #x + #y = #

Ellipses: 1x2 + ry2 + #x + #y = #

Ellipses also are horizontally stretched or compressed circles.

(r > 0)

Ellipses

EllipsesGiven two fixed points (called foci),

F2F1

EllipsesGiven two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.

F2F1

F2F1


F2F1

P Q

R

( If P, Q, and R are anypoints on an ellipse,


F2F1

P Q

R

p1

p2

( If P, Q, and R are anypoints on an ellipse, thenp1 + p2


F2F1

P Q

R

p1

p2


= q1 + q2

q1

q2


F2F1

P Q

R

p1

p2


= q1 + q2

= r1 + r2

q1

q2

r2r1


F2F1

P Q

R

p1

p2


= q1 + q2

= r1 + r2

= a constant )

q1

q2

r2r1


F2F1

P Q

R

p1

p2


= q1 + q2

= r1 + r2

= a constant )

q1

q2

r2r1

Ellipses

An ellipse also has a center (h, k );

(h, k) (h, k)

Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.

F2F1

P Q

R

p1

p2


= q1 + q2

= r1 + r2

= a constant )

q1

q2

r2r1

Ellipses

An ellipse also has a center (h, k ); it has two axes, the semi-major (long)

(h, k)

Semi Major axis

(h, k)


Semi Major axis

F2F1

P Q

R

p1

p2


= q1 + q2

= r1 + r2

= a constant )

q1

q2

r2r1

Ellipses

An ellipse also has a center (h, k ); it has two axes, the semi-major (long) and the semi-minor (short) axes.

(h, k)

Semi Major axis

(h, k)

Semi Minor axis


Semi Major axis

Semi Minor axis

These semi-axes correspond to the important radii of the ellipse.

Ellipses

These semi-axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius

Ellipses

x-radius

x-radius

y-radius

These semi-axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius and the vertical length the y-radius.

Ellipses

x-radius

x-radius

y-radius


Ellipses

x-radius

The general equation for ellipses is Ax2 + By2 + Cx + Dy = E where A and B are the same sign but different numbers.

x-radius

y-radiusy-radius


Ellipses

x-radius

The general equation for ellipses is Ax2 + By2 + Cx + Dy = E where A and B are the same sign but different numbers. Using completing the square, such equations may be transformed into the standard form of ellipses below.

x-radius

y-radiusy-radius

(x – h)2 (y – k)2

a2 b2

Ellipses

+ = 1

The Standard Form (of Ellipses)

(x – h)2 (y – k)2

a2 b2

Ellipses

+ = 1 This has to be 1.


(x – h)2 (y – k)2

a2 b2

(h, k) is the center.

Ellipses



(x – h)2 (y – k)2

a2 b2

x-radius = a


Ellipses



(x – h)2 (y – k)2

a2 b2

x-radius = a y-radius = b


Ellipses



(x – h)2 (y – k)2

a2 b2



Ellipses


Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2

42 22+ = 1


(x – h)2 (y – k)2

a2 b2



Ellipses



42 22+ = 1

The center is (3, –1).


(x – h)2 (y – k)2

a2 b2



Ellipses



42 22+ = 1

The center is (3, –1).The x-radius is 4.


(x – h)2 (y – k)2

a2 b2



Ellipses



42 22+ = 1

The center is (3, –1).The x-radius is 4.The y-radius is 2.


(x – h)2 (y – k)2

a2 b2



Ellipses


(3, -1)


42 22+ = 1

The center is (3, –1).The x-radius is 4.The y-radius is 2.


(x – h)2 (y – k)2

a2 b2



Ellipses


(3, -1) (7, -1)


42 22+ = 1

The center is (3, –1).The x-radius is 4.The y-radius is 2. So the right point is (7, –1),


(x – h)2 (y – k)2

a2 b2



Ellipses


(3, -1) (7, -1)

(3, 1)


42 22+ = 1

The center is (3, –1).The x-radius is 4.The y-radius is 2. So the right point is (7, –1), the top point is (3, 1),


(x – h)2 (y – k)2

a2 b2



Ellipses


(3, -1) (7, -1)(-1, -1)

(3, -3)

(3, 1)


42 22+ = 1

The center is (3, –1).The x-radius is 4.The y-radius is 2. So the right point is (7, –1), the top point is (3, 1), the left and bottom points are (1, –1) and (3, –3).


(x – h)2 (y – k)2

a2 b2



Ellipses


(3, -1) (7, -1)(-1, -1)

(3, -3)

(3, 1)


42 22+ = 1

The center is (3, –1).The x-radius is 4.The y-radius is 2. So the right point is (7, –1), the top point is (3, 1), the left and bottom points are (–1, –1) and (3, –3).


Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center and the x&y radii.Draw and label the top, bottom, right, left most points.

Ellipses


Group the x’s and the y’s:

Ellipses


Group the x’s and the y’s:9x2 – 18x + 4y2 – 16y = 11

Ellipses


Group the x’s and the y’s:9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients9(x2 – 2x ) + 4(y2 – 4y ) = 11

Ellipses


Group the x’s and the y’s:9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square

Ellipses


Group the x’s and the y’s:9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11

Ellipses


Group the x’s and the y’s:9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 +9

Ellipses


Group the x’s and the y’s:9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 +9 +16

Ellipses


Group the x’s and the y’s:9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16

Ellipses



Ellipses

9(x – 1)2 + 4(y – 2)2 = 36

9(x – 1)2 4(y – 2)2

36 36



+ = 1

Ellipses

9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1

9(x – 1)2 4(y – 2)2

36 4 36 9



+ = 1

Ellipses


9(x – 1)2 4(y – 2)2

36 4 36 9



+ = 1

(x – 1)2 (y – 2)2

22 32 + = 1

Ellipses


9(x – 1)2 4(y – 2)2

36 4 36 9



+ = 1

(x – 1)2 (y – 2)2

22 32 + = 1

Ellipses


Hence, Center: (1, 2), x-radius is 2, y-radius is 3.

9(x – 1)2 4(y – 2)2

36 4 36 9



+ = 1

(x – 1)2 (y – 2)2

22 32 + = 1

Ellipses


Hence, Center: (1, 2), x-radius is 2, y-radius is 3.

(-1, 2) (3, 2)

(1, 5)

(1, -1)

(1, 2)

Conic SectionsRecall that after dividing the equations of ellipsesAx2 + By2 + Cx + Dy = E by A, we obtain 1x2 + ry2 + #x + #y = #, with r > 0.

Conic SectionsRecall that after dividing the equations of ellipsesAx2 + By2 + Cx + Dy = E by A, we obtain 1x2 + ry2 + #x + #y = #, with r > 0.The number r controls the compression or extension factor along the vertical or the y-direction of the circles.


Let’s use 1x2 + ry2 = 1,ellipses centered at (0,0) as an example.



r = 1

1x2 + 1y2 = 1

11



r = 1

1x2 + 1y2 = 1

1x2 + y2 = 114

r = 1/4

1 1

21



r = 1

1x2 + 1y2 = 1

1x2 + y2 = 114

1x2 + y2 = 119

r = 1/9

r = 1/4

111

3

21



r = 1

1x2 + 1y2 = 1

1x2 + y2 = 114

1x2 + y2 = 119

1x2 + 4y2 = 11x2 + 9y2 = 1

r = 4

r = 1/9

r = 1/4

r = 9

11 111

3

2

1/2 11/3



r = 1

1x2 + 1y2 = 1

1x2 + y2 = 114

1x2 + y2 = 119

1x2 + 4y2 = 11x2 + 9y2 = 1

r = 4

r = 1/9

r = 1/4

r = 9

11 111

3

2

1/2 11/3

Ex. Verify that for 1x2 + ry2 = 1 the y-radius is 1/√r,i.e. the vertical rescale-factor is 1/√r (from the circle).

Ellipses

Ellipses

2.6ellipses x

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