We will discuss about the major and minor axes of the ellipse along with the examples.

Definition of the major axis of the ellipse:

The line-segment joining the vertices of an ellipse is called its Major Axis.

The major axis is the longest diameter of an ellipse.

Suppose the equation of the ellipse be \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 then, from the above figure we observe that the line-segment AA’ is the major axis along the x-axis of the ellipse and it’s length = 2a.

Therefore, the distance AA' = 2a.

Definition of the
minor axis of the ellipse:

The shortest diameter of an ellipse is the minor axis.

Suppose the equation of the ellipse be \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 then, putting x = 0 in equation we get, y = ± b. Therefore, from the above figure we observe that the ellipse intersects y-axis at B (0, b) and B’ (0, - b). The line segment BB’ is called the minor Axis of the ellipse. The minor axis of the ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 is along the y-axis and its length = 2b.

Therefore, the distance BB' = 2b.

Solved examples to find the major and minor axes of an ellipse:

**1.** Find the lengths of the major and minor
axes of the ellipse 3x^2 + 2y^2 = 6.

**Solution:**

The given equation of the ellipse is 3x\(^{2}\) + 2y\(^{2}\) = 6.

Now dividing both sides by 6, of the above equation we get,

\(\frac{x^{2}}{2}\) + \(\frac{y^{2}}{3}\) = 1 ………….. (i)

This equation is of the form \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a\(^{2}\) > b\(^{2}\)), where a\(^{2}\) = 2 i.e., a = √2 and b\(^{2}\) = 3 i.e., b = √3.

Clearly, a < b, so the major axis = 2b = 2√3 and the minor axis = 2a = 2√2.

**2.
**Find the lengths of the major and minor axes of the ellipse 9x\(^{2}\) + 25y\(^{2}\) - 225 = 0.

**Solution:**

The given equation of the ellipse is 9x\(^{2}\) + 25y\(^{2}\) - 225 = 0.

Now form the above equation we get,

3x\(^{2}\) + 2y\(^{2}\) = 225

Now dividing both sides by 225, we get

\(\frac{x^{2}}{25}\) + \(\frac{y^{2}}{9}\) = 1 ………….. (i)

Comparing the above equation \(\frac{x^{2}}{25}\) + \(\frac{y^{2}}{9}\) = 1 with the standard equation of ellipse \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1 (a\(^{2}\) > b\(^{2}\)) we get,

a\(^{2}\) = 25 ⇒ a = 5 and b\(^{2}\) = 9 ⇒ b = 3.

Clearly, the centre of the ellipse (i) is at the origin and its major and minor axes are along x and y-axes respectively.

Therefore,
the length of its major axis = 2a =
2** ∙ **5 = 10 units and the length of minor axis = 2b = 2** ∙ **3 = 6 units.

**● ****The Ellipse**

**Definition of Ellipse****Standard Equation of an Ellipse****Two Foci and Two Directrices of the Ellipse****Vertex of the Ellipse****Centre of the Ellipse****Major and Minor Axes of the Ellipse****Latus Rectum of the Ellipse****Position of a Point with respect to the Ellipse****Ellipse Formulae****Focal Distance of a Point on the Ellipse****Problems on Ellipse**

**11 and 12 Grade Math**__From Major and Minor Axes of the Ellipse____ to HOME PAGE__

**Didn't find what you were looking for? Or want to know more information
about Math Only Math.
Use this Google Search to find what you need.**

## New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.