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I can't understand eccentricity of conic sections intuitively ,is there an elegant way to put it ?

Eccentricity is given by

$e$=$\sqrt{(1-\frac{b²}{a²})}$

But why ?

Why does it matter that we take the ratio of two random lines?

What does it say about the curve ?Eccentricity

Why does it matter that we take the ratio of two distances mentioned in the picture ,?

Harry Case
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  • It's not just "two random lines". This is one definition of a conic section. (Not everyone will define it this way, but some do.) Choose a point to be the focus, choose a line to be the directrix. choose a ratio. Now look at any other point in the plane and measure the distances to the focus and directrix, in that order. Is the ratio of the two distances what you chose? If so, your point is on the curve. If not, it's not on the curve. – David K Oct 31 '21 at 12:54
  • Your definition of eccentricity in the first part of the question, by the way, seems to come from a different definition of a conic than the definition in the second part of the question. Are you studying from one source or jumping around between sources? – David K Oct 31 '21 at 12:59

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