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Vertex of Ellipse
Vertex of ellipse is the corner points of the ellipse at which it takes the maximum turn. The major axis cuts the ellipse at two points called the vertices of the ellipse, and the minor axis cuts the ellipse at the two covertices of the ellipse. The midpoint of the two vertices of the ellipse is the center of the ellipse.
The vertex of ellipse, foci of the ellipse, the center of the ellipse, all lie on the major axis of the ellipse. Let us learn more about the vertex of ellipse, its properties, with the help of examples, FAQs.
1.  What Is Vertex Of Ellipse? 
2.  Vertex And Covertex Of Ellipse 
3.  Terms Related To Vertex Of Ellipse 
4.  Examples On Vertex Of Ellipse 
5.  Practice Questions 
6.  FAQs On Vertex Of Ellipse 
What Is Vertex of Ellipse?
The vertex of ellipse is a point at which it makes its maximum turn. A vertex of ellipse is the point of intersection of the ellipse with its axis of symmetry. The ellipse intersects its axis of symmetry at two distinct points, and hence an ellipse has two vertices.The vertex of ellipse is also the point of intersection of the line which is passing through the foci of the ellipse and is cutting the ellipse at two distinct points.
Equation of Ellipse: \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\)
Vertices of Ellipse: (a, 0), (a, 0)
The major axis of the ellipse cuts the ellipse at two distinct points called the vertices of the ellipse. The line perpendicular to the major axis of the ellipse is the minor axis, and it cuts the ellipse at two distinct points called the covertices of the ellipse. The midpoint of the vertices of the ellipse, or the covertices of an ellipse is the center of the ellipse. The endpoints of the major axis are the vertices, and the endpoints of the minor axis are the covertices.
Vertex And Covertex Of Ellipse
The two standard forms of equations of an ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\), and \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\). These two standard forms of equations of an ellipse are based on their orientations, and each of the ellipses has different set of axis and vertices of the ellipse.
These two standard forms of equations have a different sets of vertices and covertices. The major axis of ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) is the xaxis with its vertices as (+a, 0), and the minor axis is the yaxis with its covertices (0, +b). Also for the ellipse \(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\) the major axis is the yaxis with its vertices as (0, +a), and the minor axis is the xaxis with its covertices (+b, 0).
Ellipse Equation  Major Axis  Vertices  Minor Axis  Covertices 

\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\)  xaxis  (+a, 0)  yaxis  (0, +b) 
\(\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1\)  yaxis  (0, +a)  xaxis  (+b, 0) 
Terms Related To Vertices Of Ellipse
The following terms help in a better understanding of the definition and properties of the vertex of the ellipse.
 Foci of Ellipse: The ellipse has two foci and the sum of the distances of any point on the ellipse from these two foci is a constant value. The foci of the ellipse are represented as (c, 0), and (c, 0). The midpoint of the foci is the center of the ellipse, and the distance between the two foci is 2c.
 Major Axis: The line which cuts the ellipse into two equal halves at its vertices is the major axis of the ellipse. The major axis passes through the foci of the ellipse, its center, and the vertices. For an ellipse having the equation \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) the coordinates of the vertices is (a, 0), (a, 0), and the length of the major axis is 2a units.
 Minor Axis: The minor axis of an ellipse is perpendicular to the major axis of the ellipse. The length of minor axis of ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) is 2b units. The endpoints of the minor axis of the ellipse is (a, 0), and (a, 0).
 Center of Ellipse: The point of intersection of the major axis and minor axis of an ellipse is the center of the ellipse. The center of ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) is the the origin (0. 0) of the coordinate axis. The center is the midpoint of the foci of ellipse, and also the midpoint of the vertices of the ellipse.
 Directrix of Ellipse: The directrix helps in giving the basic definition of the ellipse. The ratio of the distance of any point on the ellipse from the foci of ellipse and the directrix of the ellipse is lesser than 1. The directrix is perpendicular to the axis of the ellipse. The equation of the directrix of ellipse having the equation \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) is x = +a/e.
 Latus Rectum of Ellipse: The focal chord perpendicular to the major axis of the ellipse is the latus rectum of the ellipse. The ellipse has two foci and hence there are two latus rectums of an ellipse. The ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\) has the length of the latus rectum as 2b^{2}/a.
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Examples of Vertex of Ellipse

Example 1: Find the vertices of ellipse having the equation \(\dfrac{x^2}{36} + \dfrac{y^2}{25}\) = 1.
Solution:
The given equation of the ellipse is \(\dfrac{x^2}{36} + \dfrac{y^2}{25}\) = 1.
Comparing this with the standard equation of the ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}\) = 1 we have \(a^2\) = 36, \(b^2\) = 25.
Hence we have a = 6, and b = 5.
From this we can derive that the vertex of the ellipse is (+a, 0) = (+6, 0).
Therefore, the vertices of the ellipse are (+6, 0), (6, 0).

Example 2: Find the equation of an ellipse having an eccentricity of 2/3, and the vertices of the ellipse is (+6, 0).
Solution:
Given that the eccentricity e = 2/3, and the vertex of the ellipse is (+a, 0) = (+6, 0).
Here we have a =6, and we can apply this in the formula c = ae.
c = 6 × 2/3 = 4
b^{2} = a^{2}  c^{2}
b^{2} = 6^{2}  4^{2}
b^{2} = 36  16
b^{2} = 20
The required equation of the ellipse is \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}\) = 1
\(\dfrac{x^2}{36} + \dfrac{y^2}{20}\) = 1
Therefore the required equation of the ellipse is \(\dfrac{x^2}{36} + \dfrac{y^2}{20}\) = 1.
FAQs on Vertex of Ellipse
What Is Vertex Of Ellipse?
The vertex of ellipse are the points on the major axis of the ellipse where the major axis cuts the ellipse. The ellipse has two vertices. The minor axis also cuts the ellipse at two points which are referred as the covertices of the ellipse. The two vertices of ellipse having the equation \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}\) = 1 is (+a, 0), and (a, 0).
How Do You Find Vertex Of Ellipse?
The vertices of the ellipse are the points where the major axis cuts the ellipse. The ellipse \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}\) = 1 has the major axis as the xaxis and the vertices of the ellipse are (+a, 0), and (a, 0).
What Is The Formula For Vertex Of Ellipse?
The coordinates of vertices of an ellipse having the equation \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}\) = 1 is (+a, 0), and (a, 0).
Where Do You Use Vertex Of Ellipse?
The vertex of the ellipse can be used to find the length of the major axis, to find the center of the ellipse, and to find the end limit points of the ellipse. For an ellipse having the standard equation \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2}\) = 1, the vertices of ellipse are (+a, 0), (a, 0), the length of the major axis is 2a, and the center of the ellipse is (0, 0).
How Many Vertices Does The Ellipse Have?
The ellipse has two vertices on the major axis of the ellipse. Also the ellipse has two covertices where the minor axis cuts the ellipse at two distinct points.
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