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 linesegment 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 linesegment AA’ is the major axis along the xaxis 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 yaxis 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 yaxis 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 yaxes respectively.
Therefore, the length of its major axis = 2a = 2 ∙ 5 = 10 units and the length of minor axis = 2b = 2 ∙ 3 = 6 units.
`11 and 12 Grade Math
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