### Concept (1)

An ellipse is an important conic section and is formed by intersecting a cone with a plane that does not go through the vertex of a cone. The ellipse is defined by two points, each called a focus. From any point on the ellipse, the sum of the distances to the focus points is constant. The position of the foci determine the shape of the ellipse. The ellipse is related to the other conic sections and a circle is actually a special case of an ellipse.

### Sample Problems (20)

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Give an equation for an ellipse centered at the origin 10 units wide and 6 units tall?

Foci?
###### Problem 1
How to find the equation of an ellipse centered at the origin.
 (x + 2)² + (y − 3)² = 1 4 25
a) Graph
b) foci
###### Problem 2
How to transform the graph of a transformed ellipse and find vertices.

Find the equation for an ellipse with foci (±6,0) and co-vertices at (0,±8).

###### Problem 3
How to find the equation of an ellipse given co-vertices and foci.

Given 4x² + y² + 24x − 4y + 36 = 0.

Find the center and length of the major and minor axis.
###### Problem 4
How to find the equation of an ellipse by completing the square.
###### Problem 5
Writing the equation of an ellipse from a vertex and covertex, centered at the origin.
###### Problem 6
How to find the foci of an ellipse that is centered at the origin.
###### Problem 7
Finding the foci of an ellipse from general form of the equation.
###### Problem 8
Using key features of an ellipse centered at the origin to write the equation in standard form.
###### Problem 9
Write the equation for an ellipse not centered at the origin from the length of the axes and the center.
###### Problem 10
Using the length of the axes and the center to write the equation for an ellipse not centered at the origin.
###### Problem 11
Writing the ellipse equation from vertex and covertex when not centered at the origin.
###### Problem 12
Applying ellipses to the design of "whispering galleries".
###### Problem 13
Writing the equations for ellipses centered at the origin from the graphs.
###### Problem 14
Writing the equation for an ellipse from co-vertices and foci, centered at the origin.
###### Problem 15
Writing the equation for an ellipse from a graph.
###### Problem 16
Translations on an ellipse that is not centered at the origin.
###### Problem 17
Graphing an ellipse by turning general form into standard form.
###### Problem 18
Writing an equation for the elliptical orbit of a planet and interpreting eccentricity.
###### Problem 19
Using eccentricity of an ellipse to find a planet's closest and farthest distance from the sun.
###### Problem 20
Word problem involving maximum elliptical area.