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) = \(\sqrt{1 – {a^2\over b^2}}\)

Also Read : Different Types of Ellipse Equations and Graph

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

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