What is a set of all points in a plane such that the difference of each point of the set from the two fixed points is constant?

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• 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

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In Mathematics we often say "the set of all points that ... ".
What does it mean?

	A set is just a collection of things with some common property.
	When we collect ALL points that share a property we can end up with a line, a surface or other interesting thing.

Points can make a line

"the set of all points on a plane that are a fixed distance from a central point".

So, just a few points start to look like a circle, but when we collect ALL the points we will actually have a circle.

Try drawing one yourself (move any point):

images/geom-locus.js

(Note: the points are drawn as dots so you can see them,
but they really should have no size at all)

Surface

Imagine this happening in 3D space: all the points that are a fixed distance from a center make a sphere!

The idea of "the set of all points that ..." is used so much it has a name: Locus.

A Locus is a set of points that share a property.

So, a circle is "the locus of points on a plane that are a fixed distance from the center".

Note: "Locus" usually means that the points make a continuous curve or surface.

Example: An ellipse is the locus of points whose distance from two fixed points add up to a constant.

So, no matter where we are on the ellipse, we can add up the distance to point "F" and to point "G" and it will always be the same result.

(The points "F" and "G" are called the foci of the ellipse)

The idea of "Locus" can be used to create some weird and wonderful shapes!

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Copyright © 2022 Rod Pierce

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.

12

13

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.