Hyperbola Vs Parabola: What’s the difference?

Math, as a subject contains different formulas and processes to reach the solution, it can be challenging for students to understand the basic difference between hyperbola and parabola. While the formula and pictures help them know what is a hyperbola and parabola, it is crucial for them to understand the concept in a deeper sense.

In such situations, real-life examples along with basic differences help them understand it better. With such differences, they get to understand different formulas and properties of a parabola as well as a hyperbola. Students also explore the meaning, making it significant for their studies. Check out how these two differ and how you can implement this information in treatment. 

Meanings Explored

While the theoretical and practical application is important to teach the difference between hyperbola and parabola, it is crucial for students to first understand their meanings. The meaning helps them understand how and why the formula is generated for hyperbola and parabola, respectively.

1. Hyperbola

Have you seen the lamp right beside the bed? As you turn it, it displays two shades on the upper and bottom parts. That shade is often hyperbolic.

The curve formed when the plane cuts almost parallel to the axis is called a hyperbola. Due to the numerous angles between the axis and the plane, no two hyperbolas are exactly the same in shape. The closest points on the two arms are referred to as “vertices,” and the line segment that joins the arms is referred to as the “major axis.”

The formula XY=1 yields the hyperbola.

The two arms of the curve, also known as the branches, become parallel to one another in a parabola. The two arms or curves of a hyperbola do not become parallel. The major axis’ midway is where a hyperbola’s center is located. The real-life applications of hyperbola include guitar and monitor screens.

2. Parabola

Have you seen the thrill rainbows create? Each rainbow you see forms a parabolic shape.

When the plane cuts parallel to the cone side, a parabola is a result. An “axis of symmetry” in a parabola is a line that runs through the focus and is perpendicular to the directrix. It is referred to as the “vertex” when the “axis of symmetry” intersects the parabola. Given that they are all sliced at the same angle, parabolas all have the same shape. It is distinguished by “1” eccentricity. 

The formula y=X² yields the parabola.

A parabola is a set of points in a plane that are equally spaced apart from the directrix, or a specified straight line, and equally spaced apart from the focus, or a fixed point. There are numerous practical uses for parabolas. They are employed in the development of car headlight reflectors, telescopes, and satellite dishes.

Features of Hyperbola

• Hyperbola can be generally described as the distance between the present in a plane to two fixed points which is a positive constant.
• Hyperbola has two foci and two directrices.
• It is not necessary for hyperbolas to have any particular shape. They can be in different shapes.
• The curves of hyperbolas open more widely than that of parabolas.
• Hyperbola is a result of the intersection of the plane and the cone, only with the plane at an orientation that is not parallel to the side of the cone.
• Hyperbola is generally a disconnected curve with only a single branch.
• Hyperbola consists of two asymptotes.
• The arms existing in any hyperbola are not parallel to each other.
• The eccentricity of hyperbola has a value greater than 1.
• The two curves of the hyperbola face each other while open on opposing sides.

Features of Parabola

• A parabola can generally be described as a set of points in a plane that are at equal distances, either from a straight line or directrix and focus.
• A parabola consists of a single focus and directrix.
• A parabola is generated when a plane cuts off a conical surface that is parallel to the side of the cone.
• Parabola is generally connected to that only has a single branch.
• Parabola consists of only one curve.
• As compared to a hyperbola, a parabola has no asymptotes.
• It is only a parabola when the two arms existing are parallel to each other.
• The curves of a parabola open less widely as compared to that of a hyperbola.
• Parabola has an eccentricity value of 1. 
• The parabolas are of the same shape no matter their size.

Other significant differences between hyperbola and parabola

Parabolas and hyperbolas are produced using various techniques, despite the fact that both figures are conic sections. When a plane that is parallel to a cone’s side passes through the cone, a parabola is created. When a plane and a cone connect, a hyperbola is produced, but the plane’s orientation is not parallel to the cone’s side.

The characteristics of the parabola and hyperbola as conic sections also vary. A path or locus of points at a specific distance from a fixed line and a fixed point can be used to describe any conic section. The eccentricity, which is equal to the distance from a point on the curve to a fixed point divided by the perpendicular distance from the same point on the curve to the fixed line, can be used to distinguish between these two types of conic sections.

A hyperbola has an eccentricity greater than one compared to a parabola, which has an eccentricity of one.

These two curves’ graphs are also slightly different. Hyperbolas have a larger opening than parabolas. The most obvious difference between their graphs is that a hyperbola has two curves that mirror each other and open in opposing directions. A parabola, on the other hand, has only one curve.

Table of Comparison

Summing up,

As you’ve browsed through the differences between hyperbola and parabola, use them to check basic differences, mathematical formulas, and real-life applications. These differences are sure to create a comprehensive learning environment by offering straightforward information on the concept. Teachers and parents can also opt for different books, online games, quotes, and classroom activities to offer a diverse learning platform for students.

Manpreet Singh

An engineer, Maths expert, Online Tutor and animal rights activist. In more than 5+ years of my online teaching experience, I closely worked with many students struggling with dyscalculia and dyslexia. With the years passing, I learned that not much effort being put into the awareness of this learning disorder. Students with dyscalculia often misunderstood for having  just a simple math fear. This is still an underresearched and understudied subject. I am also the founder of Smartynote -‘The notepad app for dyslexia’,