IXL - Write equations of ellipses in standard form using properties (Algebra 2 practice)

Write the equation in standard form for the ellipse with vertices

(

)

and

(

–

)

and co-vertices

(

–

)

and

(

)

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question

Write the equation in standard form for the ellipse with vertices

(

)

and

(

–

)

and co-vertices

(

–

)

and

(

)

key idea

An ellipse that has a horizontal orientation is wider than it is tall. An ellipse that has a vertical orientation is taller than it is wide.

solution

To write the equation in standard form, find the center

(

)

Then, find the values of

and

and determine if the ellipse has a horizontal or vertical orientation.

Find the center.

The vertices are the endpoints of the major axis and the co-vertices are the endpoints of the minor axis. The center is the midpoint of both the major and minor axes. To find the center, plug the vertices or co-vertices into the midpoint formula.

Plug in

and

–

Plug in

and

–

(

)

The center is

(

)

which is the origin.

Find

and

and determine the orientation.

The distance from the center to each vertex is

Since the center

(

)

and the vertex

(

)

have the same

-coordinate,

is the absolute value of the difference in their

-coordinates,

which is

–

Also, since the vertices are on the

-axis,

the major axis is vertical and the ellipse has vertical orientation.

The distance from the center to each co-vertex is

Since the center

(

)

and the co-vertex

(

)

have the same

-coordinate,

is the absolute value of the difference in their

-coordinates,

which is

–

So,

and

Write the equation in standard form.

The equation of the ellipse in standard form is:

(

–

)

(

–

)

Ellipse with a vertical orientation

(

–

)

(

–

)

Plug in

and

Plug in

and

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Not feeling ready yet? These can help:

FF.4Write equations of ellipses in standard form from graphs

EE.4Write equations of circles in standard form using properties

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