What is the intersection of a cone and a plane parallel to a line along the side of a cone? Show an equation or example to support your answer.
The intersection of a right circular cone and a plane parallel to a side of the cone is a parabola. (In the degenerate case that the plane contains the side of the cone, the intersection is a line.)
Let A be the vertex of the cone. Let B,E,C,D lie on a circle formed by a plane perpendicular to the axis of the cone. Then the plane parallel to the line of the cone intersects the circle at E,P, and D with P the vertex of the parabola. The diameter BC meets the chord (in the case of a right circular cone a diameter) ED at its midpoint M.
A plane perpendicular to the axis of the cone containing P is drawn with radius r and diameter PK. Let EM=DM=x and let PM=y.
Then `BM=2ysintheta ` (Since triangle PBM is isosceles.)
`CM=2r ` (PMCK is a parallelogram.)
`EM*DM=BM*CM ` (BC and ED are intersecting chords in a circle.)
Substituting we get:
This is the formula for a parabola in standard form. For a given cone, ` r ` and `theta ` are constant while x and y depend on the choice of the circle BEC.