Here you will learn how to find the coordinates of the vertices and center of ellipse formula with examples.
Let’s begin –
Vertices and Center of Ellipse Coordinates
(i) For the ellipse \(x^2\over a^2\) + \(y^2\over b^2\) = 1, a > b
The coordinates of vertices are (a, 0) and (-a, 0).
And the coordinates of center is (0, 0)
(ii) For the ellipse \(x^2\over a^2\) + \(y^2\over b^2\) = 1, a < b
The coordinates of vertices are (0, b) and (0, -b).
And the coordinates of center is (0, 0)
Also Read : Different Types of Ellipse Equations and Graph
Example : For the given ellipses, find the coordinates of vertices and center
(i) \(16x^2 + 25y^2\) = 400
(ii) \(x^2 + 4y^2 – 2x\) = 0
Solution :
(i) We have,
\(16x^2 + 25y^2\) = 400 \(\implies\) \(x^2\over 25\) + \(y^2\over 16\) = 1,
where \(a^2\) = 25 and \(b^2\) = 16 i.e. a = 5 and b = 4
Clearly a > b,
Center of ellipse is (0, 0)
And Vertices of ellipse is (a, 0) and (-a, 0).
\(\implies\) (5, 0) and (-5, 0)
(ii) We have,
\(x^2 + 4y^2 – 2x\) = 0
\(\implies\) \((x – 1)^2\) + 4\((y – 0)^2\) = 1
\(\implies\) \((x – 1)^2\over 1^2\) + \((y – 0)^2\over (1/2)^2\) = 1
Here, a = 1 and b = 1/2
Clearly a > b,
Coordinates of Center of the ellipse is (h, k) i.e. (1, 0)
And Coordinates of Vertices are (a + h, k) and (-a + h, k)
\(\implies\) (1 + 1, 0) and (-1 + 1, 0) = (2, 0) and (0, 0)
Note : For the ellipse \((x – h)^2\over a^2\) + \((y – k)^2\over b^2\) = 1 with center (h. k),
(i) For ellipse a > b,
The coordinates of vertices are (a + h, k) and (-a + h, k).
And Coordinates of Center is (h, k).
(ii) For ellipse a < b,
The coordinates of vertices are (h, b + k) and (h, -b + k).
And Coordinates of Center is (h, k).
