How to Find Equation of Ellipse when given Foci

Ellipse is an important topic in the conic section. 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 called foci of the ellipse. In this article, we will learn how to find the equation of ellipse when given foci.

The centre of the ellipse is the midpoint of the line segment joining the foci of the ellipse. The major axis is the line segment passing through the foci of the ellipse. The line segment passing through the centre and perpendicular to the major axis is called the minor axis. The endpoints of the major axis are called the vertices of the ellipse. The distance between the foci is denoted by 2c. So the length of the semi-major axis is a and the semi-minor axis is b. The length of the major axis is denoted by 2a and the minor axis is denoted by 2b.

The relation between the semi-major axis, semi-minor axis and the distance of the focus from the centre of the ellipse is given by the equation c = √(a2 – b2).

The standard equation of ellipse is given by (x2/a2) + (y2/b2) = 1.

The foci always lie on the major axis. The major axis can be known by finding the intercepts on the axes of symmetry, i.e, the major axis is along the x-axis if the coefficient of x2 has the larger denominator and it is along the y-axis if the coefficient of y2 has the larger denominator.

Steps to find the Equation of the Ellipse.

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

2. If the coordinates of the vertices are (±a, 0) and foci is (±c, 0), then the major axis is parallel to x axis. Then use the equation x2a2+y2b2=1\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1.

3. If the coordinates of the vertices are (0, ±a) and foci is (0,±c), then the major axis is parallel to y axis. Then use the equation x2b2+y2a2=1\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1.

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

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

Solved Examples

Example 1.

Find the equation of an ellipse with vertices (0, ±8) and foci (0, ±4).

Solution:

Given vertices (0, ±8) and foci (0, ±4).

Here a = 8 and c = 4

c2 = a2 – b2

So b2 = a2 – c2

= 82 – 42

= 64 – 16

= 48

We use the equation x2b2+y2a2=1\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}} = 1.

So (x2/48) + y2/64 = 1 is the required equation.

Example 2.

Find the equation of the ellipse that has vertices (± 13, 0) and foci are (± 5, 0).

Solution:

Given vertices (± 13, 0) and foci are (± 5, 0).

Vertices are in x-axis. So the equation will be of the form (x2/a2) + (y2/b2) = 1.

Here a = 13 and c = 5

b2 = a2 – c2

b2 = 132 – 52

b2 = 144

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