Vertices and Center of Ellipse Coordinates

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).

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