Foci of Ellipse Formula and Coordinates

Here you will learn how to find the coordinates of the foci of ellipse formula with examples.

Let’s begin –

Foci of Ellipse Formula and Coordinates

(i) For the ellipse \(x^2\over a^2\) + \(y^2\over b^2\) = 1, a > b

The coordinates of foci are (ae, 0) and (-ae, 0)

(ii) For the ellipse \(x^2\over a^2\) + \(y^2\over b^2\) = 1, a < b

The coordinates of foci are (0, be) and (0, -be)

Also Read : Different Types of Ellipse Equations and Graph

Example : For the given ellipses, find the coordinates of foci.

(i)  \(16x^2 + 25y^2\) = 400

(ii)  \(x^2 + 4y^2 – 2x\) = 0

Solution :

(i)  We have,

\(16x^2 + 25y^2\) = 400 \(\implies\) \(x^2\over 25\) + \(y^2\over 16\),

where \(a^2\) = 25 and \(b^2\) = 16 i.e. a = 5 and b = 4

Clearly a > b,

The eccentricity of ellipse (e) = \(\sqrt{1 – {b^2\over a^2}}\)

e = \(\sqrt{1 – 16/25}\) = \(3\over 5\)

Therefore, the coordinates of foci are (ae, 0) and (-ae, 0)

\(\implies\) (3, 0) and (-3, 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,

The eccentricity of ellipse (e) = \(\sqrt{1 – {b^2\over a^2}}\)

e = \(\sqrt{1 – 1/4}\) = \(\sqrt{3}\over 2\)

Since, Center of the above ellipse (h, k) i.e. (1, 0)

Therefore, the coordinates of foci are (ae + h, k) and (-ae + h, k)

\(\implies\) (\(\sqrt{3}\over 2\) + 1, 0) and (-\(\sqrt{3}\over 2\) + 1, 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 foci are (ae + h, k) and (-ae + h, k).

(ii) For ellipse a < b,

The coordinates of foci are (h, be + k) and (h, -be + k).

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