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# Major Axis|Definition & Meaning

## Definition of Major Axis

**Major axis** is the **longer axis** of the **ellipse** that connects the two **farthest points** of its **perimeter** while passing through its center.

When we talk from the perspective of geometry, an **ellipse** is a type of standard shape which is **two-dimensional** and which can be described by its two **axes.** The **elliptical shape** is generally created when a **plane** intersects a **conic** at a tilted angle with the base of the cone. All ellipses have **two axes. **

**Figure 1: Axes of an Ellipse**

The **major** **axis** is the ellipse’s **longest** **diameter** (denoted as **‘a’** by convention), which goes through the **ellipse’s center** from one end to the other. The **minor** **axis** (denoted as **‘b’** by convention) is the ellipse’s **smallest** **diameter,** passing through the center at its **narrowest** **point.**

## Explanation of Major Axis of Ellipse

A **line segment** connecting the elliptical shape’s farther vertices is termed a **major axis.** It can also be defined as the **longest** one of the two **diameters** of the ellipse.

Let us say that we have an ellipse with the **center at C,** as shown in **figure** **2**. Now we can see the **four vertices** of the ellipse at points marked as A, A’, B, and B.’

**Figure 2: Mathematical Form of Major Axis of an Ellipse**

The line segments joining the **opposite vertices** and passing through center C are called the **axes** of this **ellipse.** The line segment BB’ joining the closest vertices, that is B and B’, is known as the **minor axis** of the ellipse. While the line segment AA’ joining the farthest vertices, that is A and A’, is termed as the **major axis** of the ellipse.

In the above diagram, the distance of the **farthest vertices,** that is, A and A’, from the center C is equal to ‘a’. Therefore, the distance from A to A’ will be a + a = 2a which is the **length** of the **line segment** AA’ and the **length** of the **major axis.**

## Mathematical Form of Major Axis of an Ellipse

Mathematically, the equation of an ellipse can be written in two-dimensional space in the **following standard form:**

\[ \dfrac{x^2}{a^2} \ + \ \dfrac{y^2}{b^2} \ = \ 1 \]

In the above formula, the **two dimensional Cartesian coordinates** have been represented by **x and y**. **‘a’ and ‘b’ defines the distances between the center of the ellipse and its vertices**. Since the ellipse may be oriented in any direction or at any rotation angle, we may not explicitly say which one of these lengths is a part of the **minor axis** or **major axis.**

It is a common practice first to solve the equation given in the question such that it resembles the **standard form** (as given in the equation above). Once we have this standard form, we can evaluate the **values of a and b**. After that, we can compare which one of these is a larger distance. Suppose **if a > b**, the length of the **major axis will be 2a,** and the length of the **minor axis will be 2b**.

Similarly, **if a < b**, then the length of the minor axis will be 2a, and the length of the **major axis will be 2b.** We can see this process in further detail under the **numerical problems section** of this article.

## Examples of Major Axis of Ellipse

There are many **real-life examples** of the major axis of an ellipse. The **following** paragraphs detail only two of such examples.

The first example is the one of an **American football.** The ball, as shown in figure 3, is **elliptical.** We can mathematically define its dimensions by specifying the **two distances** of its **major** and **minor axes.** The distance of the **two notches,** as shown in the figure, represents the major axis, while the diameter of the ball right in the **middle section** represents its minor axis length.

**Figure 3: Football Example of Major Axis of an Ellipse**

The second example is also a very familiar one: the **solar system.** The following figure shows the path of our home planet around the Sun. We know that all the **planets circulating **the **Sun** follow an elliptical path. This **elliptical path** means that during the whole year-long cycle of Earth, our globe is not at a constant distance from the Sun. Rather it keeps changing, and this change creates **seasons on Earth.** The **major axis** of the Earth’s orbit is **marked** in the following **figure.**

**Figure 4: Solar System Example of Major Axis of an Ellipse**

## Numerical Examples of Major Axis

### Example 1

Given the **following equation** of an ellipse, find both **(minor as well as major)** axes of this ellipse.

9x$\mathsf{^2}$ + 4y$\mathsf{^2}$ = 36

### Solution

Given:

9x$\mathsf{^2}$ + 4y$\mathsf{^2}$ = 36

**Converting** into **standard form:**

\[ \dfrac{ 9 x^{ 2 } }{ 36 } \ + \ \dfrac{ 4 y^{ 2 } }{ 36 } \ = \ \dfrac{ 36 }{ 36 } \]

\[ \dfrac{ x^{ 2 } }{ 4 } \ + \ \dfrac{ y^{ 2 } }{ 9 } \ = \ 1 \]

**Comparing** with the **standard form,** we have the following:

a$\mathsf{^2}$ = 4

a = 2

b$\mathsf{^2}$ = 9

b = 3

Comparing ‘a’ and ‘b,’ since **b > a,** ‘b’ is the part of the **major axis** while ‘a’ is the part of the **minor axis.**

Length of Major Axis = 2b = 2(3) = 6

Length of Minor Axis = 2a = 2(2) = 4

### Example 2

Given the **following equation** of an ellipse, find both **(minor as well as major)** axes of this ellipse.

16x$\mathsf{^2}$+ 25y$\mathsf{^2}$ = 400

### Solution

Given:

16x$\mathsf{^2}$+ 25y$\mathsf{^2}$ = 400

**Converting** in **standard form:**

\[ \dfrac{ 16 x^{ 2 } }{ 400 } \ + \ \dfrac{ 25 y^{ 2 } }{ 400 } \ = \ \dfrac{ 400 }{ 400 } \]

\[ \dfrac{ x^{ 2 } }{ 25 } \ + \ \dfrac{ y^{ 2 } }{ 16 } \ = \ 1 \]

**Comparing** with the **standard form,** we have the following:

a$\mathsf{^2}$ = 25

**a = 5**

b$\mathsf{^2}$ = 16

**b = 4**

Comparing ‘a’ and ‘b,’ since **a > b,** ‘a’ is the part of the **major axis** while ‘b’ is the part of the **minor axis.**

Length of Major Axis = 2a = 2(5) = 10

Length of Minor Axis = 2b = 2(4) = 8

*All images/mathematical drawings were created with GeoGebra.*