Question 1038815
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It is not sufficient for the plane to intersect the axis of the cone.  The plane must be parallel to a generator of the cone.  If the angle between the axis and the plane is greater than the angle between a generator of the cone and the axis, then the intersection of the plane and the cone is an ellipse (or in the extreme case, a circle).  If the angle between the axis and the plane is less than the angle between the axis and a generator of the cone, the intersection is a hyperbola.  Obviously, since all generators of a cone intersect the axis at the apex, any plane parallel to a generator must, perforce, also intersect the axis.


If the plane does NOT intersect the axis, then you have a hyperbola with symmetric branches.  But the plane intersects the axis for all other conic sections including non-symmetrical hyperbolae. 


As the plane moves toward the apex, the parabola becomes narrower.  That is to say that the parameter *[tex \Large a] in the vertex form of the equation of a parabola, *[tex \Large y\ =\ a(x\ -\ h)^2\ +\ k], becomes larger.  When the plane is coincident with the apex, the intersection becomes a straight line that is one of the generators of the cone.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
<img src="http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg">
*[tex \Large \ \
*[tex \LARGE \ \ \ \ \ \ \ \ \ \  

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