Question 1038815: Imagine a plane and a cone intersecting to form a parabola.
A. Explain why the plane has to intersect the axis of the cone.please
B. Imagine the plane moving so that it keeps the same angle, but it's point of intersection with the axis moves in the direction of the apex of the cone and eventually passes through the apex. Describe what happens to the parabola and write equations that describe how the parabola changes.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
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  in the vertex form of the equation of a parabola, ^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
My calculator said it, I believe it, that settles it
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