## The Differences Between Parabola vs Hyperbola

## History of Hyperbola

The word “Hyperbola” was derived from the Greek the word which means “overthrown” or “excessive” ‘and from English “Hyperbole” was derived .hyperbolae were discovered in his investigation of the problem of doubling the cube but those were called the section of obtuse cones. The term hyperbola is believed to have been coined by Apollonius of Perga (c. 262–c. 190 BC) in his definitive work on the conic sections.

## History of parabola

The word ‘parabola’ refers to the parallelism of the conic section and the tangent of the conic mantle. Also the parable has been derived from the Greek ‘parabole’.

The parabola can be seen as an ellipse with one focus in infinity. This means that a parallel light bundle in a parabolic mirror will come together at one point. It had been told that Archimedes did use a parabolic mirror in warfare. It was during the siege of Syracuse (214 – 212 BC) by the Romans, that Archimedes constructed reflecting plates in about the form of a parabola. These plates were used to coverage the sunlight onto the Roman ships and put them in fire. Though this event is discussed by some historians, recently the feasibility of Archimedes’ plan has been proved 2).

Other technological parabola shaped objects are the parabolic microphone and the parabolic antenna, used to focus sound and electromagnetic waves, respectively.

## What is Conic?

In mathematics, a **conic section** (or simply **conic**) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic sections are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga’s systematic work on their properties.

**Parabola vs Hyperbola**

First of all, the question which arises is that what is parabola and Hyperbola. I will explain the differences between both by mathematical explanation or explain in an easy way by which not only mathematic but also common man will understand. Before starting the discussion, we must know what the cone is. A cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface pointed towards the top. The pointed end of the cone is called the **apex**, whereas the flat surface is called the Base. These are the two different sections of a cone.

A solid figure of the cone when it is cut by a plane the section which we get is called a conic section. Conic sections could be circles, ellipses, hyperbolas, and parabolas depending upon the angle of intersection between axis of the cone and the pane.

The Arms or branches of curves are open of hyperbolas and parabolas which means they will continue to infinity; they are not close curves like a circle or an ellipse.

## What is Hyperbola?

The line portion joining the vertices of a hyperbola is known as the transverse pivot and generally signified as 2 a. The midpoint of this line fragment is the hyperbolas inside. The transverse pivot of a hyperbola is its line of balance. A different line of balance is opposite to the transverse hub and is known as the conjugate hub. Its length is indicated as 2 b. The accepted condition of a hyperbola in the Cartesian facilitate framework is written in the structure.

**Hyperbola is given by the condition XY=1**

At the point when the distinction of separations between a lot of focuses present in a plane to two fixed foci or focuses is a positive steady, it is known as a hyperbola.

The absolute value of the difference of the distances from any point of a hyperbola to its foci is constant.|r1−r2|=2a, where r1, r2 are the distances from an arbitrary point P(x,y) of the hyperbola to the foci F1 and F2, a is the transverse semi-axis of the hyperbola.

### Equations of the asymptotes of a hyperbola

**y=±abx**

**Relationship between semi-axes of a hyperbola and its focal distance**

**c ^{2}=a^{2}+b^{2}**

Where c is half the focal distance, is the transverse semi-axis of the hyperbola, b is the conjugate semi-axis.

## What is Parabola?

If we talk about the mathematic a **parabola** is a plane curve that is mirror-symmetrical and is approximately U-shaped. Every point of which has the property that the distance to a fixed point (called the focus of the parabola) is equal to the distance to a straight line (the directrix of the parabola). The line perpendicular to the directrix and passing through the focus is called the axis of symmetry. Where the parabola meets its pivot of evenness is known as the “vertex” and where the parabola is most forcefully bent. The separation between the vertex and the center, estimated along with the pivot of balance, is the “central length”. The separation between the concentrations to the directrix is known as the central parameter and signified by p.

Parabola | Hyperbola |
---|---|

When a set of points in a plane are equidistant from a given directrix or a straight line and the focus then it is called a parabola. | When the difference of distances between a set of points present in a plane to two fixed points is a positive constant, it is called a hyperbola. |

All parabolas are of the same shape irrespective of the size. | All hyperbolas are of different shapes. |

The two arms in a parabola become parallel to each other. | They do not become parallel. |

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