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 !

  • $\begingroup$ The two definitions are equivalent. It’s a useful exercise to prove this for yourself. $\endgroup$ – amd Feb 28 '18 at 19:08
  • $\begingroup$ There's a connection between the geometric eccentricity and the eccentricity of orbits, which I've written up here: mathhelpboards.com/math-notes-49/…. $\endgroup$ – Adrian Keister Feb 28 '18 at 19:10

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 $${0-{de^2\over e^2-1} \over {de\over e+1}-{de^2\over e^2-1}} = e.$$


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