Major and Minor Axes of the Ellipse

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 x2a2 + y2b2 = 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 x2a2 + y2b2 = 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 x2a2 + y2b2 = 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 3x2 + 2y2 = 6.

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

x22 + y23 = 1 ………….. (i)

This equation is of the form x2a2 + y2b2 = 1 (a2 > b2), where a2 = 2 i.e., a = √2 and b2 = 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 9x2 + 25y2 - 225 = 0.

Solution:

The given equation of the ellipse is 9x2 + 25y2 - 225 = 0.

Now form the above equation we get,

3x2 + 2y2 = 225

Now dividing both sides by 225, we get

x225 + y29 = 1 ………….. (i)

Comparing the above equation x225 + y29 = 1 with the standard equation of ellipse x2a2 + y2b2 = 1 (a2 > b2) we get,

a2 = 25 ⇒ a = 5 and b2 = 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 = 25 = 10 units and the length of minor axis = 2b = 23 = 6   units.

● The Ellipse





11 and 12 Grade Math 

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