Ellipses Part 1

Circle/Ellipse Quiz: March 9Midterm: March 11

Definition:

An ellipse is the set of all points in a plane such that the sum of the distance from P to two fixed point (F1, F2) call foci is constant.

Ellipses can have a Horizontal Major Axis or a Vertical Major Axis:

Major Axis Is the longer axis

of the ellipse The endpoints of

the major axis are called VERTICES.

The coordinates for your foci are on the major axis.

Minor Axis Is the shorter

axis of the ellipse The endpoints of

the minor axis are called CO-VERTICES.

Horizontal Major Axis

2 2

2 21

x y

a b

Vertical Major Axis

2 2

2 21

x y

b a

Note:

When the bigger number is under the x-term, the major axis will be on the x-axis (or parallel to it if translated)

When the bigger number is under the y-term, the major axis will be on the y-axis (or parallel to it if translated).

Note

The length of the major axis is 2a The length of the minor axis is 2b The foci are always on the major

axis The following are always true for

ellipses: a2 > b2

a2 – b2 = c2

Note

“a” is your vertices coordinate “b” is your co-vertices coordinate “c” is your foci coordinate

Example 1:

Identify the vertices, co-vertices, and foci of the ellipse with equation 9x2 + y2 = 36

Example 2:

Write the standard form equation for an ellipse with foci of (0, -4) and (0, 4) and with minor axis of 6.

Example 3:

Write the standard form equation for an ellipse with foci of (-8,0) and (8,0) with major axis of 20.

You Try:

Identify the vertices, co-vertices, and foci of the ellipse with equation2 2

181 4

x y

You Try:

Write the standard equation for an ellipse with foci (-12, 0) and (12, 0) and with a major axis of 26.

Homework

M3 book (GREEN) P. 188 #1-15

Ellipses Part 2

Circle/Ellipse Quiz: March 9Midterm: March 11

Translated Ellipses

Horizontal Major Axis:

Vertical Major Axis:

1

2

2

2

2

b

ky

a

hx

2 2

2 21

x h y k

b a

Example 3:

Write the standard equation for an ellipse with its center at (2,-4) and with a horizontal major axis of 10 and minor axis of 6. Identify vertices and co-vertices.

**Use the value of a to find the vertices, and the value of b to find the co-vertices

You Try:

Write the standard equation for an ellipse with its center at (-1,-2) and with a vertical major axis of 8 and minor axis of 4.

Give the vertices and co-vertices

Example 4:

An ellipse is defined by 4x2 + y2 + 24x – 4y + 36 = 0. Write the standard equation, and identify the coordinates of its center, vertices, co-vertices, and foci. Sketch the graph.

The Graph:

You Try:

An ellipse is defined by the equation 4x2 + 25y2 – 24x + 200y + 336=0. Write the standard equation and identify the coordinates of the center, vertices, co-vertices, and foci.

The Graph:

Homework:

Re-work all the Ellipse examples from class You NEED to practice, practice, practice

Ellipse WS