**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**.

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