Parabola and hyperbola are two different sections of a cone. This article will explain the difference between them.

First of all, when a solid figure, like a cone, is cut by a plane, the section obtained is called a conic section. Depending upon the angle of intersection that is between the plan and axis of the cone, Conic sections will be circles, ellipses, hyperbolas, and parabolas. As well as being open curves, parabolas and hyperbolas are also closed curves, but they are not closed curves like a circle or ellipse.

**Parabola**

When a plane cuts parallel to the cone side a curve is obtained that is called a parabola. A line that passes through the focus and is perpendicular to the directrix is called an axis of symmetry in a parabola. As parabolas are cut at a specific angle, they are all shaped identically. When the parabola intersects the point on the axis of symmetry, it is known as the vertex. Despite having the same shape, they can have different sizes due to the eccentricity of “1.”

Hyperbola

Hyperbolas are formed when the distance between two fixed foci or points in a plane is positive.

Hyperbolas are curves obtained by cutting almost parallel to axes. Because of the angles between the axis and the plane, hyperbolas do not have the same shape. In geometry, “vertices” are the points closest to each other, while “major axes” are the lines connecting them.

As a parabola curve expands, its arms, sometimes referred to as branches, become parallel. In a hyperbola, two curves do not become parallel. Hyperbola’s center is the midpoint of the major axis.

Hyperbola is given by the equation [katex] XY=1 [/katex]

**What is Hyperbola?**

When a cone is cut by a plan the intersection is not a circle, ellipse, or point. Hyperbolas are a common term for mathematics. For example, hyperbolas are used to describe the path of a projectile under the effect of gravity, the shape of a satellite dish, and the orbit of a comet around the sun. There are many types of hyperbolas, But they all have two basic qualities: they have two arms that extend from a central point to outward, and increasingly move away from the center. Hyperbolas can be open or closed, it depends on whether their arms extend to infinity or not.

Difference between parabola and hyperbola

Basis of comparison | Parabola | Hyperbola |

Definition | Parabolas are loci where points are equally distanced from a focus or directrix. | A hyperbola consists of points that have the same distance from two foci |

Intersection of Plan | It is ideal if the plane intersects the cone parallel | It is parallel to the congruent height of the double cone. |

General Equation | The general equation of the parabola is [katex] y = ax², a ≠ 0[/katex] | The general equation of the hyperbola is [katex] x²/a² – y²/b² = 1 [/katex] |

Eccentricity | The non-negative eccentricity of a parabola is one. | The non-negative eccentricity e of a hyperbola is greater than one. |

Shape | Parabolas are open curves with one directrix. | There are two foci and two directives on the hyperbola, which is an open curve with two branches. |