CBSE Class 11 Mathematics Conic Sections Notes

Key Concepts

1. Let l  be a fixed vertical line and m be another line intersecting it at a fixed point V and inclined to it at and angle a

On rotating the line m around the line l in such a way that the angle a remains constant a surface is generated is a double-napped right circular hollow cone.

2. The point V is called the vertex; the line l is the axis of the cone. The rotating line m is called a generator of the cone. The vertex separates the cone into two parts called nappes.

3. The sections obtained by cutting a double napped cone with a plane are called conic sections. Conic sections are of two types (i) degenerate conics (ii) non degenerate conics.

4. If the cone is cut at its vertex by the plane then degenerate conics are obtained.

5. If the cone is cut at the nappes by the plane then non degenerate conics are obtained.

6. Degenerate conics are point, line and double lines.

7. Circle, parabola, ellipse and hyperbola are degenerate conics.

8. When the plane cuts the nappes (other than the vertex) of the cone,degenerate conics are obtained.

(a) When β = 90o , the section is a circle.

The plane cuts the cone horizontally.

(b) When α < β < 90o, the section is an ellipse.

The plane cuts one part of the cone in an inclined manner

(c) When β = α; the section is a parabola.

The plane cuts the cone in such a way that it is parallel to a generator

(d) When 0 ≤  β < α; the plane cuts through both the nappes the curve of intersection is a hyperbola.

The plane cuts both parts of the cone.

9. When the plane cuts at the vertex of the cone, we have the following different cases:

(a) When α <β ≤ 90⁰, then the section is a point.

(b) When β = α, the plane contains a generator of the cone and the section is a straight line.

It is the degenerated case of parabola.

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