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.

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embizze | High School Teacher | (Level 1) Educator Emeritus

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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:

`x*x=2ysintheta*2r `

`x^2=4rysintheta `

`y=x^2/(4rsintheta) `

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.

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