# 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)

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).