A conic section is a curve obtained when a plane intersects a cone at some specific angle. There are three types of conic sections – ellipse, parabola, and hyperbola.

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An ellipse is a planar curve that has two focal points, and somewhat resembles a circle. However, the parabola and hyperbola are confusing sections.

**Parabola vs Hyperbola**

The difference between a parabola and a hyperbola is that the parabola is a single open curve with eccentricity one, whereas a hyperbola has two curves with an eccentricity greater than one.

A parabola is a single open curve that extends till infinity. It is U-shaped and has one focus and one directrix.

A hyperbola is an open curve having two unconnected branches. It has two foci and two directrices, one for each branch.

## Comparison Table

Parameter Of Comparison | Parabola | Hyperbola |
---|---|---|

Definition | A parabola is a locus of the points that have equal distance from a focus and a directrix. | A hyperbola is a locus of the points that have a constant difference from two foci. |

Shape | The parabola is an open curve that has one focus and one directrix. | The hyperbola is an open curve with two branches that has two foci and two directrices. |

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

Intersection of Plane | The intersection of the plane is parallel (ideal case) to the slant height of the cone. | The intersection of the plane is parallel (ideal case) to the perpendicular height of the double cone. |

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

## What is Parabola?

A parabola is the locus of all the points that are equidistant from a point and a line. This point is called the focus, and this line is called the directrix.

A parabola is formed when a plane intersects a cone in a direction parallel (ideal case) to its slant height.

The general equation of a parabola is given as

y = ax² , a ≠ 0

The value of a determines the shape of the curve.

If a > 0, the mouth of the parabola opens to the top.

If a < 0, the mouth of the parabola opens to the bottom.

The focus of the above parabola is (0, 1/4a). The directrix is (-1/4a).

However, when a=1, the parabola is called a unit parabola.

A parabola has an eccentricity of one.

A parabola is symmetric about its axis. At an infinite distance, the curves appear as parallel lines.

## What is Hyperbola?

A hyperbola is the locus of all the points that have a constant difference from two distinct points. These points are called the foci of the hyperbola.

A hyperbola is formed when a solid plane intersects a cone in a direction parallel to its perpendicular height.

The general equation of a hyperbola is given as

(x-α) ²/a² – (y-β)²/b² = 1

The foci of the above hyperbola are ( α ± sqrt( a²+b²), β).

The vertices are (±a, β).

A hyperbola has an eccentricity greater than one.

A hyperbola has two axes of symmetry. These are the transverse axis and the conjugate axis.

**Main Differences Between Parabola and Hyperbola**

A parabola and a hyperbola are conic sections. They have different shapes and properties.

The main differences between the two are :

- A parabola is a locus of all the points that have equal distance from a focus and a directrix. On the other hand, a hyperbola is a locus of all the points for which the difference in distance between two foci is constant.
- A parabola is an open curve having one focus and directrix, whereas a hyperbola is an open curve with two branches having two foci and directrices.
- The eccentricity of a parabola is one, whereas the eccentricity of a hyperbola is greater than one.
- A parabola is formed when the plane intersects a cone along its slant height. On the other hand, a hyperbola is formed when the plane intersects a cone along its perpendicular height.
- The equation for a parabola is y = ax². On the other hand, the equation for a hyperbola is x²/a² – y²/b² = 1.

## References

- https://www.osapublishing.org/abstract.cfm?uri=ao-54-24-7148
- https://asmedigitalcollection.asme.org/appliedmechanics/article-abstract/68/4/537/449711