Which is the definition of eccentricity of an ellipse

Question

I've been doing some research into conic sections and I'm getting confused as to what the actual definition of the eccentricity of the conic section is.

Now from what I understand the eccentricity is defined as the ratio between the lengths FP and PD where F is the focus, P is the point and D is the perpendicular length to the directrix. I thought this applied to all conic sections.

I've also seen elsewhere as it being defined (for the ellipse) as: e=distance of focus from center/length of semi major axis

I'm sure it's a case of symmetry about the minor axis but I'm not sure where to start. Thank you for your help !

The two definitions are equivalent. It’s a useful exercise to prove this for yourself. — amd, Feb 28 '18 at 19:08
There's a connection between the geometric eccentricity and the eccentricity of orbits, which I've written up here: https://mathhelpboards.com/math-notes-49/connection-between-angle-conic-section-energy-corresponding-orbit-4469.html. — Adrian Keister, Feb 28 '18 at 19:10
Does this answer your question? Confusion with the eccentricity of ellipse — David K, Oct 31 '21 at 14:52

score 1 · Answer 1 · answered Feb 28 '18 at 21:35

The two definitions are equivalent. Consider w.l.o.g. an ellipse with focus at the origin, directrix $x=d>0$ and eccentricity $e$. The focus-directrix equation of this ellipse is $$x^2+y^2 = e^2(x-d)^2.$$ The center and right-hand vertex of the ellipse are easily found to be at $\left({de^2\over e^2-1},0\right)$ and $\left({de\over e+1},0\right)$, respectively. The ratio of the semi-focal distance to the semi-major axis length is therefore $${0-{de^2\over e^2-1} \over {de\over e+1}-{de^2\over e^2-1}} = e.$$

Which is the definition of eccentricity of an ellipse

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