# Difference Between Hyperbola vs Parabola

If you are starting out your math journey, you will learn several interesting topics. But then, many students have a phobia of the subject. So, this could be the reason you did not grasp the difference between hyperbola and parabola when it was taught in class.

The good thing is that this guide will walk you through the subject. However, before getting into the nitty-gritty, you must remember that the concepts have to do with graphical math. Now, you need to understand the definitions.

## Definition of Hyperbola

A hyperbola is the locus of all the points that have a constant difference from the two distinct points. The term for the point is loci of a hyperbola. One can also describe it as the shape formed whenever a plane cuts through a cone in a direction parallel to its height.

It has an eccentricity that is more than ONE. With two axes of symmetry lying opposite each other, the two parts of a hyperbola are formed. The two axes are known as transverse and conjugate axes.

The general equation of hyperbola is (x-a)^{2}/a^{2} – (y- β)^{2}/b^{2} = 1

Therefore, the loci is {a ± sqrt (a^{2}+b^{2}), β]

Despite its common use in the classroom, the term also has its application in the real world. For instance, astronauts depend on it for determining the accuracy of spaceship and satellite launches.

In other words, it gives them an accurate path to launch the spaceship or satellite when it leaves the earth. The same applies to radio systems, as scientists use it to determine the areas that radio signals should cover.

## Definition of Parabola

A parabola is the locus of all the points that are equidistant from the point and a line. It is a common term in graphs and linear equations. While that point is known as a focus, the line is called the directrix. One can also see it as the shape formed when a plane cuts through a cone in a way that it is parallel to the slant height.

To determine a parabola, the equation is always given as y = ax^{2}. However, the condition is that “a” is NOT equal to zero. In other words, a ≠ 0. If the value of “a” is given, then it is easy to determine the shape of the curve in the equation.

There are other interesting facts. Well, if the value of “a” is greater than ZERO, it means that the parabola’s mouth will open to the top. On the other hand, the same will open to the bottom of the value of “a” if it is less than ZERO.

Before getting into the difference between parabola and hyperbola, keep in mind that the former also has its real-world applications. It is often applied in using satellites to direct signals to the appropriate antennas. In a similar vein, engineers apply it during bridge construction for suspending bridges. Some table fans and standing fans use this principle too.

## Main Differences Between Hyperbola vs Parabola

The table below clarifies the hyperbola vs parabola concept.

Basis of Comparison | Hyperbola | Parabola |

Definition | The locus of points that have fixed disparity from the two foci | The locus of the points that have equal distance from the focus |

Shape formed | An open, two-branched curve with two foci and two directrices | An open curve with a focus and a directrix |

Shape formed at intersection | Is parallel to the perpendicular height of the double cone | Is parallel to the slant height of the cone |

General equation | The general equation is given as (x-a)^{2}/a^{2} – (y- β)^{2}/b^{2} = 1 | The general equation is given as y = ax^{2} |

Number of asymptotes | Two | Nil |

Nature of shape | Different shapes | The same shape, no matter the size |

## Difference Between Hyperbola and Parabola: Conclusion

To wrap up this parabola vs hyperbola tutorial, it is important to point out that the two shapes are always obtained whenever a cone-shaped object is split into two.

It is also noteworthy that the two shapes are both open curves, meaning that their arms and branches have no limits. In conclusion, you have learned the most critical distinctions between these two math concepts.