3 ellipses

Ellipses

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

P Q

R

If P, Q, and R are anypoints on a ellipse,


F2F1

P Q

R

p1

p2

If P, Q, and R are anypoints on a 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 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 has a center (h, k ); it has two axes, the major (long)

(h, k)(h, k)


Major axis

Major axis

F2F1

P Q

R

p1

p2


= q1 + q2

= r1 + r2

= a constant

q1

q2

r2r1

Ellipses

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

(h, k)Major axis

Minor axis

(h, k)

Major axis

Minor axis


These axes correspond to the important radii of the ellipse.Ellipses

These 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 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 transform to 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, major and minor axis.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)

Ellipses

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