Conic Section

Conic section is a closed or open curve obtained by the intersection of a cone and a plane. Let $O$ be a fixed point. From $O$, draw a fixed line $OC$. Now, let a line $OA$ rotate around $OC$ such that $\angle AOC$ is always contant. Then, the surface generated by rotating the line $OA$ around $OC$ is known as right cone.

The point $O$ is the vertex. The fixed line $OC$ is the axis and the rotating line $OA$ is the generator. $\angle AOC$ is known as the semi vertical angle. Consider another cone $OA’B’$ such that it is symmetrical to the cone $OAB$ about $OC’$ opposite to $OC$. Then, $ABOA’B’$ is said to be the double right cone.

If a plane cuts the cone, then a curve is obtained. That curve is called conic section. The nature of the curve is determined by the position of the cutting plane. Different curves or conic sections obtained when a cone is intersected by a plane in different positions are given below.

Intersection of a Cone and a Plane

If a plane intersects a cone perpendicular to the axis, then the section is a circle.

If a plane intersects a cone at a given angle with the axis greater than the semi vertical angle, then the section is an ellipse.

If a plane, not passing through the vertex, intersects a cone parallel to the generator of the cone, then the section is a parabola.

If a plane intersects the double right cone such that the angle between the axis and the plane be less than the semi vertical angle, then the section is a hyperbola.

Hence, conic section is the locus of a point which moves in a plane in such a way that the ratio of its distance from a fixed point to its distance from a fixed straight line is constant.

The fixed point is called the focus, the fixed straight line its directrix, and the constant ratio the eccentricity (denoted by $e$). The straight line passing through the focus and perpendicular to the directrix is called the axis. The intersection of the curve and the axis is called the vertex.

1. A conic section in which the value of the eccentricity is unity i.e. $e=1$ is known as the parabola.
2. A conic section in which the value of the eccentricity is less than $1$ i.e. $e<1$ is known as the ellipse.
3. A conic section in which the value of the eccentricity is greater than $1$ i.e. $e>1$ is known as the hyperbola.