# How to Find Equation of Ellipse with Foci and Major Axis

When we consider the conic section, an ellipse is an important topic. It is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. These fixed points are known as foci of the ellipse. The major axis is the line segment passing through the foci of the ellipse. In this article, we will learn how to find the equation of ellipse with foci and major axis.

The distance between the foci is denoted by 2c. The length of the major axis is denoted by 2a and the minor axis is denoted by 2b.

## Steps to Find the Equation of the Ellipse with Foci and Major Axis

1. Find whether the major axis is on the x-axis or y-axis.

2. If major axis is on x-axis then use the equation $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$.

3. If major axis is on y-axis then use the equation $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1$.

4. Find ‘a’ from the length of the major axis. Length of major axis = 2a

5. Using the equation c2 = (a2 – b2), find b2.

6. Substitute the values of a2 and b2 in the standard form.

The standard form of the equation of an ellipse with center (h,k) and major axis parallel to x axis is

((x-h)2 /a2)+((y-k)2/b2) = 1

When a>b

Major axis length = 2a

Coordinates of the vertices are (h±a,k)

Minor axis length is 2b

Coordinates of covertices are (h,k±b)

Coordinates of foci are (h±c,k). Also c2= a2-b2

## Solved Examples

Example 1:

Find the equation of the ellipse, whose length of the major axis is 20 and foci are (0, ± 5).

Solution:

Given the major axis is 20 and foci are (0, ± 5).

Here the foci are on the y-axis, so the major axis is along the y-axis.

So the equation of the ellipse is

x2/b2 + y2/a2 = 1

2a = 20

a = 20/2 = 10

a2 = 100

c = 5

c2 = a2 – b2

b2 = a2 – c2 = 102 – 52 = 75

So (x2/75) + y2/100 = 1 is the required equation.

Example 2:

Find the equation of the ellipse whose length of the major axis is 26 and foci (± 5, 0)

Solution:

Given the major axis is 26 and foci are (± 5,0).

Here the foci are on the x-axis, so the major axis is along the x-axis.

So the equation of the ellipse is x2/a2 + y2/b2 = 1

2a = 26

a = 26/2 = 13

a2 = 169

c = 5

c2 = a2 – b2

b2 = a2 – c2 = 132 – 52 = 144

So (x2/169) + y2/144 = 1 is the required equation.