# Eccentricity of Ellipse – Formula and Examples

Here you will learn what is the eccentricity of ellipse formula and how to find eccentricity with examples.

Let’s begin –

## Eccentricity of Ellipse Formula

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

we have,  $$b^2$$ = $$a^2(1 – e^2)$$

$$\implies$$  $$e^2$$ = 1 – $$b^2\over a^2$$

$$\implies$$  eccentricity (e) = $$\sqrt{1 – {b^2\over a^2}}$$

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

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

Example : For the given ellipses, find the eccentricity.

(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,

Therefore, the eccentricity of ellipse (e) = $$\sqrt{1 – {b^2\over a^2}}$$

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

(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,

Therefore, the eccentricity of ellipse (e) = $$\sqrt{1 – {b^2\over a^2}}$$

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