MATHEMATICS S6 UNIT 8: CONICS.

Conic Sections (Conics)

Definition: Conic sections are curves formed by the intersection of a plane with a double-napped right circular cone. Depending on the angle of the intersecting plane relative to the cone, different types of curves are produced.

The Four Main Types:

1. Circle: Formed when the plane is perpendicular to the cone’s axis (parallel to the base). It’s a special case of an ellipse.
2. Ellipse: Formed when the plane intersects only one nappe of the cone and is not perpendicular to the axis, resulting in a closed curve.
3. Parabola: Formed when the plane intersects only one nappe of the cone and is parallel to exactly one of the cone’s generating lines (the lines that form the cone). It’s an open curve.
4. Hyperbola: Formed when the plane intersects both nappes of the cone, resulting in two separate, open curves.

Alternative Definition (Eccentricity): Conics can also be defined as the locus of points where the ratio of the distance from a fixed point (the focus) to a fixed line (the directrix) is a constant, called the eccentricity ($e$)

• $e=0$: Circle
• $0<e<1$: Ellipse
• $e=1$: Parabola
• $e>1$: Hyperbola

Algebraic Representation: All conic sections can be represented by a general second-degree polynomial equation in two variables:

where  A,B,C,D,E and F are real numbers and , A,B,C are not all nulls.

A conic section is the set of all points which move in a plane such that its
distance from a fixed point and a fixed straight line not containing the
fixed point are in a constant ratio.
We use the term degenerate conic sections to describe the single point,
single straight line and the term non-degenerate conic sections to
describe parabola, ellipse or hyperbola.
The three non-degenerate conics (the parabola, ellipse and hyperbola)
can be defined as the set of points P in the plane that satisfy the following
condition:
The distance from a fixed point F (called the focus of the conic) to point
variable P is a constant multiple of distance from a fixed straight line (called
its directrix) to point P. This constant multiple is called its eccentricity, ‘e’.