What is the set of all points in the plane the sum of whose distances from two fixed points?

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.

Horizontal ellipse, center at origin	[latex]\dfrac{{x}^{2}}{{a}^{2}}+\dfrac{{y}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex]
Vertical ellipse, center at origin	[latex]\dfrac{{x}^{2}}{{b}^{2}}+\dfrac{{y}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex]
Horizontal ellipse, center [latex]\left(h,k\right)[/latex]	[latex]\dfrac{{\left(x-h\right)}^{2}}{{a}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,\text{ }a>b[/latex]
Vertical ellipse, center [latex]\left(h,k\right)[/latex]	[latex]\dfrac{{\left(x-h\right)}^{2}}{{b}^{2}}+\dfrac{{\left(y-k\right)}^{2}}{{a}^{2}}=1,\text{ }a>b[/latex]

An ellipse is the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form.
When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse.
When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse.
Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci.

center of an ellipse

the midpoint of both the major and minor axes

conic section

any shape resulting from the intersection of a right circular cone with a plane

ellipse

the set of all points [latex]\left(x,y\right)[/latex] in a plane such that the sum of their distances from two fixed points is a constant

foci

plural of focus

focus (of an ellipse)

one of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point [latex]\left(x,y\right)[/latex] on the ellipse is a constant

major axis

the longer of the two axes of an ellipse

minor axis

the shorter of the two axes of an ellipse