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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 x2a2 + y2b2 = 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 co-ordinates 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 9x2 + 16y2 - 144 = 0.
Solution:
The given equation of the ellipse is 9x2 + 16y2 - 144 = 0
Now form the above equation we get,
9x2 + 16y2 = 144
Dividing both sides by 144, we get
x216 + y29 = 1
This is the form of x2a2 + y2b2 = 1, (a2 > b2), where a2 = 16 or a = 4 and b2 = 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 9x2 + 16y2 - 144 = 0 are (4, 0) and (-4, 0).
2. Find the coordinates of the vertices 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,
9x2 + 25y2 = 225
Dividing both sides by 225, we get
x225 + y29 = 1
Comparing the equation x225 + y29 = 1
with the standard equation of ellipse x2a2 + y2b2 = 1 (a2 > b2) we get,
a2 = 25 or a = 5 and b2 = 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 9x2
+ 25y2 - 225 = 0 are (5, 0) and (-5, 0).
● The Ellipse
11 and 12 Grade Math
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