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.

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

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

= q1 + q2

= r1 + r2

= q1 + q2

= r1 + r2

= a constant

= q1 + q2

= r1 + r2

= a constant

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.

= q1 + q2

= r1 + r2

= a constant

Ellipses

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

(h, k)(h, k)

Major axis

= q1 + q2

= r1 + r2

= a constant

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

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

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

Ellipses

The Standard Form (of Ellipses)

(x – h)2 (y – k)2

Ellipses

+ = 1 This has to be 1.

(x – h)2 (y – k)2

(h, k) is the center.

Ellipses

(x – h)2 (y – k)2

x-radius = a

Ellipses

(x – h)2 (y – k)2

x-radius = a y-radius = b

Ellipses

(x – h)2 (y – k)2

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

Ellipses

42 22+ = 1

The center is (3, –1).

(x – h)2 (y – k)2

Ellipses

42 22+ = 1

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

(x – h)2 (y – k)2

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

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

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

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

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

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

9(x – 1)2 + 4(y – 2)2 = 36

9(x – 1)2 4(y – 2)2

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

Ellipses

9(x – 1)2 4(y – 2)2

36 4 36 9

(x – 1)2 (y – 2)2

22 32 + = 1

Ellipses

9(x – 1)2 4(y – 2)2

36 4 36 9

(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

(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

3 ellipses

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