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.

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

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**Parabola vs Hyperbola**

**The main 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 Between Parabola and Hyperbola (in Tabular Form)

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.

## Conclusion

Conic sections comprise of ellipses, parabole and hyperbole. They are referred to as conic sections because they are derived by the intersection of a cone with a plane. Parabolas are a single infinite curve. They are the locus of points equidistant from the focus and directrix.

Hyperbolas are curves with two branches. They are the locus of points that have a constant difference in distance from two foci. The difference lies in their eccentricities. Parabolas have an eccentricity of one, while hyperbolas have an eccentricity greater than one.

Parabolas have various applications in real life. They are used in architecture, engineering, spacecraft design, reflectors, and holographic films. Hyperbolas are popular in radio engineering, satellite design, lenses, computers, and sundials. In fact, our universe is in the form of a hyperbola.

## References

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

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