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

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

Grade:12th pass

1 Answers

Arun
25758 Points
3 years ago
 

Sections of a Cone

It depends upon the location of the plane with reference to the cone and the angle made by the vertical axis and the intersecting plane, that what kind of conic section we get. 


Sections of a Cone

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

Degenerate conics

 

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