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Defining conic sections

As part of Edexcel Further Pure 1 revision, I have come across rectangular hyperbolas. I wish to look into the more fundamental definitions of hyperbolas generally, and then conic sections in general. In terms of the Edexcel A Level syllabus, it simply states the parametric Cartesian equations for parabolas, and rectangular hyperbolas, with no further explanations. So I guess this is an undergraduate question.

From what I have learnt so far there are four classes of conic section defined eitehr by how you slice two infintie cones, or using the focus & directrix distance to produce eccentricity. Circle, ellipse, parabola, hyperbola. Happy with deriving deriving directrix-focus definition of ellipse and parabola. However I am struggling with circles and hyperbolas.

Usind directrix-focus definition, a circle has e=0. Now for this to be true, then focus to point distance/point to directrix distance must =0 ie focus to point distance must be =0. For this to be the case a circle is only defined with 0 radius, yet be definition a circle has radius > 0, so how does one solve this contradiction?! Look at the link to an animation of how e affects conic section shape:
http://www.mathsisfun.com/geometry/eccentricity.html
THis shows how a circle (with e=0) is a point with no radius.

Secondly, I cannot seem to find anywhere a method of deriving the calculation of e for a hyperbola using only the directrix-focus definition. Many places show how to derive the Cartesian equation of a hyperbola (x^2/a^2 - y^2/b^2 =1) using the loci of each branch meansured form the two foci. This does not really address how to define a branch using the directrix-focus, in teh same fashion as can be done for a parabola.

If anyone is able to address my concerns for circles, and hyperbolas, that would be most helpful.
Reply 1
Avatar for ztibor
ztibor
10
Original post by Choochoo_baloo

If anyone is able to address my concerns for circles, and hyperbolas, that would be most helpful.


http://en.wikipedia.org/wiki/Eccentricity_(mathematics)
Original post by ztibor


Hello,

Had a look through Wikipedia already, it does not address my queries. I was hoping someone could explain to me my problems from first principles etc. I wish it was as simple as learning it off of wikipedia!!
Reply 3
Avatar for Hasufel
Hasufel
16
I`ve made this up for myself, so maybe it might help...

technically, a circle is an ellipse with zero eccentricity. for the ellipse, e=1−(ba)2e=\sqrt{1-(\frac{b}{a})^{2}}

and, since the nearer a gets to b or vice versa, the eccentricity gets closer to zero (the two "focii" coincide)

for a circle, the directrix becomes the tangent to the circle.
(edited 10 years ago)
cooooniccs.pdf347.4KB

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