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
11 and 12 Grade Math
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