Equation of an Ellipse with Center at the Origin

An ellipse is defined as the set of all points (x, y) in a plane so that the sum of their distances from two fixed points is constant. Each fixed point is called a focus of the ellipse. All ellipses have two lines of symmetry. The longest axis is called the major axis and the shortest axis is called the minor axis. Each extreme point of the major axis is the vertex of the ellipse and each extreme point of the minor axis is the co-vertex of the ellipse. The center of an ellipse is the midpoint of both the major and minor axes. The axes of the ellipse are perpendicular at the center. The foci are always located on the major axis.

elements of an ellipse

In this article, we will learn about the equation of the ellipse with center at the origin. We will only focus on ellipses that are positioned vertically or horizontally on the Cartesian plane.

PRECALCULUS
Parameters of horizontal ellipses centered at the origin

Relevant for

Finding the equation of the ellipse with center at the origin.

See equation

PRECALCULUS
Parameters of horizontal ellipses centered at the origin

Relevant for

Finding the equation of the ellipse with center at the origin.

See equation

Standard form of ellipses with center at the origin

Standard forms of equations give us information about the main characteristics of graphs. The main characteristics of the ellipse are its center, vertices, covertices, foci, and lengths and positions of the major axis and the minor axis. There are four variations of the standard form of an ellipse.

These variations depend first on the location of the center (at the origin or outside the origin), and then on the orientation of the ellipse (vertical or horizontal).

Equation of the horizontal ellipse

The standard form of an ellipse with center at the origin, (0, 0), and with the major axis parallel to the x-axis is:

\frac{{{x}^2}}{{{a}^2}}+\frac{{{y}^2}}{{{b}^2}}=1

where,

  • a>b
  • The length of the major axis is 2a
  • The length of the minor axis is 2b
  • The coordinates of the vertices are (\pm a, 0)
  • The coordinates of the covertices are (0, \pm b)
  • The coordinates of the foci are (\pm c, 0), where, {{c}^2}={{a}^2}-{{b}^2}
Parameters of horizontal ellipses centered at the origin

Equation of the vertical ellipse

The equation of an ellipse in its standard form that has its center at the origin, (0, 0), and in which its major axis is parallel to the y axis is:

\frac{{{x}^2}}{{{b}^2}}+\frac{{{y}^2}}{{{a}^2}}=1

where,

  • a>b
  • The major axis measures 2a
  • The minor axis measures 2b
  • The coordinates of the vertices are (0, \pm a)
  • The coordinates of the covertices are (\pm b, 0)
  • The coordinates of the foci are (0, \pm c), where, {{c}^2}={{a}^2}-{{b}^2}
Parameters of vertical ellipses centered at the origin

Writing the equation for ellipses with center at the origin using vertices and foci

To find the equation of an ellipse centered on the origin given the coordinates of the vertices and the foci, we can follow the following steps:

Step 1: Determine if the major axis is located on the x-axis or on the y axis.

1.1. If the coordinates of the vertices have the form (\pm a, 0) and the coordinates of the foci have the form (\pm c, 0), the major axis is parallel to the x axis and we use the equation \frac{{{x}^2}}{{{a}^2}}+\frac{{{y}^2}}{{{b}^2}}=1.

1.2. If the coordinates of the vertices have the form (0, \pm a) and the coordinates of the foci have the form (0, \pm c), the major axis is parallel to the y axis and we use the equation \frac{{{x}^2}}{{{b}^2}}+\frac{{{y}^2}}{{{a}^2}}=1.

Step 2: We use the equation {{c}^2}={{a}^2}-{{b}^2} along with the coordinates of the vertices and foci and we solve for {{b}^2}.

Step 3: We substitute the values of {{a}^2} and {{b}^2} into the equation obtained in step 1.


Ellipse with center at the origin – Examples with answers

The following examples put into practice what you have learned about the equation of ellipses that have their center at the origin. Go through each example to understand the process used to get the answer.

EXAMPLE 1

Find the equation of the ellipse that has vertices at (±7, 0) and foci at (±4, 0).

Solution

The foci are on the x-axis, so the major axis is on the x-axis. This means that the equation will have the following form:

\frac{{{x}^2}}{{{a}^2}}+\frac{{{y}^2}}{{{b}^2}}=1

The vertices ate (\pm 7, 0), so that a=7 and we have {{a}^2}=64.

The foci are (\pm 4, 0), so that c=4 and we have {{c}^2}=16.

Now, we use the equation {{c}^2}={{a}^2}-{{b}^2} to obtain the value of {{b}^2}. Therefore, we have:

{{c}^2}={{a}^2}-{{b}^2}

16=64-{{b}^2}

{{b}^2}=48

We have to substitute the obtained values in the standard equation. Therefore, the equation of the ellipse is:

\frac{{{x}^2}}{64}+\frac{{{y}^2}}{48}=1

EXAMPLE 2

What is the equation of the ellipse that has vertices at (0, ±9) and foci at (0, ±6)?

Solution

In this case, the foci are on the y axis. This means that the major axis is on the y axis. Therefore, the equation will have the following form:

\frac{{{x}^2}}{{{b}^2}}+\frac{{{y}^2}}{{{a}^2}}=1

The vertices are (0, \pm 9), so that a=9 and we have {{a}^2}=81.

The foci are (0, \pm 6), so that c=6 and we have {{c}^2}=64.

Using the equation {{c}^2}={{a}^2}-{{b}^2}, we can get the value of {{b}^2}. Therefore, we have:

{{c}^2}={{a}^2}-{{b}^2}

64=81-{{b}^2}

{{b}^2}=17

Using these values, we have the following equation for the ellipse:

\frac{{{x}^2}}{17}+\frac{{{y}^2}}{81}=1


Ellipse with center at the origin – Practice problems

Put into practice what you have learned to solve the following problems. Find the equations of the ellipses with the given information. If you need help with this, you can look at the solved examples above.

What is the equation of the ellipse that has the vertices (\pm8, 0) and the foci (\pm 5, 0)?

Choose an answer






What is the equation of the ellipse that has the vertices (0, \pm8) and the foci (0, \pm \sqrt{5})?

Choose an answer







See also

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