Is the set of all points in the plane the difference of whose distances from two fixed F1 and F2 is a constant?

Dear student, A hyperbola is "the set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant". The difference of the distances to any point on the hyperbola (x,y) from the two foci (c,0) and (-c,0) is a constant. That constant will be 2a. Regards.

1 Definition: An ellipse is the set of all points in a plane such that the sum of the distances from P to two fixed points (F1 and F2) called foci is constant. 9.4 Ellipse

2 A few terms: 1.Major axis: 2. Minor axis: 3. Vertices: 4.Co-vertices: 5. Foci: 6. Center: Longer axis of the ellipse Shorter axis of the ellipse Endpoints of the Major axis (a’s) Endpoints of the Minor axis (b’s) Always on the Major axis (c’s) Where the Major and Minor intersect (h,k)

3 Ellipses can have either a horizontal major axis or a vertical major axis. Horizontal Major Axis Vertical Major Axis 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). Center (h, k)

4 -The length of the major axis is _______ -The length of the minor axis is ________ -The foci are always on the __________ axis. - The following are ___________ true for ellipses: a: b: c:

5 To write an equation you need the CENTER and the a’s and b’s. Example 1: Write the standard form equation for an ellipse with foci of (0, -4) and (0, 4) and with minor axis of 6.

6 To write an equation you need the CENTER and the a’s and b’s. Example 2: Write the standard form equation for an ellipse with foci of (-8, 0) and (8,0) and with major axis of 20.

7 Example 3: Find the vertices and co-vertices. Is the major axis vertical or horizontal?

8 Example 4: Put the following equations in standard form for an ellipse. State the vertices, co-vertices, and foci. Does the ellipse have a horizontal or vertical major axis?

9 Example 5: Sketch. Label the center, foci, vertices and co-vertices.

10 Example 6: Write the standard equation for an ellipse with the given characteristics. a. Foci: (5,0) and (-5, 0 )b. Co-vertices: (0,2) and (0, -2) Vertices: (9, 0) and (-9, 0) Vertices: (3,0) and (-3, 0)

11 Example 7: Write a standard form equation for each ellipse. Identify the center, foci, vertices, and co-vertices.