We will discuss about the vertex of the ellipse along with the examples.
Definition of the vertex of the ellipse:
The vertex is the point of intersection of the line perpendicular to the directrix which passes through the focus cuts the 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 perpendicular to the directrix KZ and passing through the focus S cuts the ellipse at A and A'.
The points A and A', where the ellipse meets the line joining the foci S and S' are called the vertices of the ellipse.
Therefore, the ellipse has two vertices A and A' whose coordinates are (a, 0) and ( a, 0) respectively.
Solved examples to find the vertex of an ellipse:
1. Find the coordinates of the vertices of the ellipse 9x\(^{2}\) + 16y\(^{2}\)  144 = 0.
Solution:
The given equation of the ellipse is 9x\(^{2}\) + 16y\(^{2}\)  144 = 0
Now form the above equation we get,
9x\(^{2}\) + 16y\(^{2}\) = 144
Dividing both sides by 144, we get
\(\frac{x^{2}}{16}\) + \(\frac{y^{2}}{9}\) = 1
This is the form of \(\frac{x^{2}}{a^{2}}\) + \(\frac{y^{2}}{b^{2}}\) = 1, (a\(^{2}\) > b\(^{2}\)), where a\(^{2}\) = 16 or a = 4 and b\(^{2}\) = 9 or b = 3
We know the coordinates of the vertices are (a, 0) and (a, 0).
Therefore, the coordinates of the vertices of the ellipse 9x\(^{2}\) + 16y\(^{2}\)  144 = 0 are (4, 0) and (4, 0).
2. Find the coordinates of the vertices 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,
9x\(^{2}\) + 25y\(^{2}\) = 225
Dividing both sides by 225, we get
\(\frac{x^{2}}{25}\) + \(\frac{y^{2}}{9}\) = 1
Comparing the 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 or a = 5 and b\(^{2}\) = 9 or b = 3
We know the coordinates of the vertices are (a, 0) and (a, 0).
Therefore, the coordinates of the vertices of the ellipse 9x\(^{2}\)
+ 25y\(^{2}\)  225 = 0 are (5, 0) and (5, 0).
● The Ellipse
11 and 12 Grade Math
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