How can we obtain a line when a plane cut the cone? Explain with the help of 3-D diagram.

Let the angle between intersecting plane and the vertical axis be ∅.

If the plane cuts the one nappe of the cone, such that,

• Ɵ ellipse.

• ∅ = 90°, when the plane is perpendicular at the axis, or the angle of intersection of plane and the vertical axis is equal to 90°, the ellipse becomes the circle. Or you can say that circle is the special type of ellipse, the section is circle.

• Ɵ = ∅, when the angle between the vertical axis and the plane is equal to the vertex angle, we get an open curve called Parabola at the intersecting surface of the cone.

• If the plane cuts the two nappes of the cone, such that,

• Ɵ Hyperbola.
   

Degenerate conics

If a plane intersects the double right circular cone at the vertex of the cone, such that

• When Ɵ point .

• When Ɵ = ∅, in case of parabola, the plane contains a generator of the cone and the section is a straight line.

• When 0 ≤ Ɵ pair of intersecting straight lines