Open In App

How to find the equation of an ellipse given the foci and minor axis?

Improve
Improve
Like Article
Like
Save
Share
Report

Conic sections, commonly known as conics, are formed when a plane intersects a cone. The angle at which these sections intersect determines their shape. Conic sections are therefore classified into four types: circle, ellipse, parabola, and hyperbola. Each of these types has its own set of equations and mathematical properties. The ellipse is discussed below.

Ellipse

An ellipse is a conic section generated when a plane meets a cone at an angle (β) that is less than the right angle but greater than the angle formed at the cone’s vertex (α). To put it another way, an ellipse is generated when a plane slices a cone at an angle β such that α<β<90o.

A cone and a plane cross at an angle β that is less than the right angle but more than α to produce an ellipse, as illustrated in the diagram above.

Equation of an ellipse

  • The standard equation of an ellipse centered at (h, k) with a major axis parallel to the x-axis is given by:

\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1  ,

where the coordinates of the vertex are (h±a, 0), coordinates of co-vertex are (h, k±b) and the coordinates of foci are (h±c, k), where c2 = a2 – b2.

Horizontal ellipse

  • The standard equation of an ellipse centered at (h, k) with a major axis parallel to the y-axis is given by:

\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1

where the coordinates of the vertex are (h, k±a), coordinates of co-vertex are (h±b, k) and the coordinates of foci are (h, k±c), where c2 = a2 – b2.

Vertical ellipse

How to find the equation of an ellipse given the foci and minor axis?

Solution:

To find the equation of an ellipse, we need the values a and b. Now, we are given the foci (c) and the minor axis (b). To calculate a, use the formula c2 = a2 – b2. Substitute the values of a and b in the standard form to get the required equation.

Let us understand this method in more detail through an example.

Example: Say, an ellipse centered at origin with foci (±4, 0) and minor axis (0, ±3).

Given b = 3 and c = 4.

Put these in the formula c2 = a2 – b2 to find a.

a2 = 32 + 42

a2 = 25

a = 5

As the ellipse lies on x-axis, the equation is of the form \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1   .

So, the equation is, \frac{x^2}{25}+\frac{y^2}{9}=1   .

Similar Problems

Question 1. Find the equation of an ellipse centered at the origin with foci (±7, 0) and minor axis (0, ±5).

Solution:

Given b = 5 and c = 7.

Put these in the formula c2 = a2 – b2 to find a.

a2 = 52 + 72

a2 = 74

As the ellipse lies on x-axis, the equation is of the form \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1

So, the equation is, \frac{x^2}{74}+\frac{y^2}{25}=1

Question 2. Find the equation of an ellipse centered at the origin with foci (0, ±5) and minor axis (12, 0).

Solution:

Given b = 12 and c = 5.

Put these in the formula c2 = a2 – b2 to find a.

a2 = 52 + 122

a2 = 169

a = 13

As the ellipse lies on y-axis, the equation is of the form \frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1

So, the equation is, \frac{x^2}{144}+\frac{y^2}{169}=1

Question 3. Find the equation of an ellipse centered at (3, 2) with c = 6 and b = 8. 

Solution:

Given b = 8, c = 6, h = 3 and k = 2.

Put these in the formula c2 = a2 – b2 to find a.

a2 = 82 + 62

a2 = 100

a = 10

As the ellipse lies on x-axis, the equation is of the form \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1

So, the equation is, \frac{(x-3)^2}{100}+\frac{(y-2)^2}{64}=1

Question 4. Find the coordinates of the major axis of the ellipse with foci (0, ±5) and minor axis (12, 0).

Solution:

We have, c = 5 and b = 12.

Put these in c2 = a2 – b2 to find a.

a2 = 122 + 52

a2 = 169

a = 13

The coordinates of major axis are (0, ±13).

Question 5. Find the coordinates of the major axis of the ellipse with foci (±24, 0) and minor axis (0, 10).

Solution:

We have, c = 24 and b = 10.

Put these in c2 = a2 – b2 to find a.

a2 = 102 + 242

a2 = 676

a = 26

The coordinates of major axis are (0, ±26).



Last Updated : 21 Feb, 2023
Like Article
Save Article
Previous
Next
Share your thoughts in the comments
Similar Reads