# Parabola vs. Hyperbola: Know the Difference

By Shumaila Saeed || Published on February 12, 2024

**Parabola is a U-shaped curve; every point is equidistant from a fixed point and line. Hyperbola is an open curve formed by intersecting a cone with a plane at an angle parallel to the cone's side.**

## Key Differences

Parabolas are geometric shapes characterized by their symmetrical, U-shaped curves. Each point on a parabola is equidistant from a fixed point (focus) and a fixed straight line (directrix). Hyperbolas, in contrast, consist of two separate, mirror-image curves. They result from intersecting a cone with a plane at an angle parallel to one of the sides of the cone.

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Feb 12, 2024

In a parabola, the axis of symmetry is perpendicular to the directrix, creating its distinct U-shape. This shape is common in physics and engineering, exemplified in the trajectory of projectiles. Hyperbolas are defined by two foci and two directrices, with the curves approaching these lines asymptotically. Hyperbolas appear in navigation systems and astrophysics.

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Feb 12, 2024

Parabolas have numerous practical applications, such as in the design of satellite dishes and car headlights, where their reflective properties are essential. Hyperbolas also find real-world applications, notably in the paths of celestial bodies in certain orbital patterns and in hyperbolic geometry.

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Feb 12, 2024

In optics, a parabola reflects parallel rays to a single focus, used in telescopes and searchlights. Hyperbolas, with their two asymptotes, model radar and sonar's signal paths, illustrating their diverse utility in different scientific fields.

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Feb 12, 2024

## Comparison Chart

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### Applications

Reflective properties in optics and engineering

Models in navigation, astrophysics, and hyperbolic geometry

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Feb 12, 2024

## Parabola and Hyperbola Definitions

#### Parabola

In mathematics, a parabola represents a quadratic graph, characterized by its U-shape.

The graph of y=x2 is a simple parabola.

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#### Hyperbola

A hyperbola is a geometric figure with two focal points, where the difference in distances to any point on the curve is constant.

Hyperbolas model the behavior of radar waves.

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Jan 14, 2024

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#### Parabola

A parabola is defined geometrically as a set of points equidistant from a point (focus) and a line (directrix).

The shape of a satellite dish is a parabola to focus signals at its receiver.

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Jan 14, 2024

#### Hyperbola

A hyperbola is an open curve formed by intersecting a cone with a plane at a shallow angle.

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Jan 14, 2024

#### Parabola

A parabola is a symmetric curve made by cutting a cone parallel to its side.

The trajectory of a basketball shot is a perfect parabola.

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Jan 14, 2024

#### Hyperbola

Hyperbolas are used in navigation and astronomy to describe certain orbital paths and signal ranges.

GPS technology utilizes hyperbolic equations for pinpointing locations.

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#### Parabola

In physics, a parabola describes the path of projectiles under uniform gravity, neglecting air resistance.

A thrown ball follows a parabolic path.

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Jan 14, 2024

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#### Hyperbola

A hyperbola is defined by its two asymptotes and two disconnected branches.

The paths of some comets around the sun are hyperbolic.

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#### Parabola

A parabola is a mirror-symmetrical plane curve, common in algebra and geometry.

The path of light in a parabolic mirror is redirected to its focus.

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#### Hyperbola

In mathematics, a hyperbola represents a type of smooth curve lying in a plane.

The hyperbola is crucial in defining hyperbolic functions in trigonometry.

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#### Parabola

A plane curve formed by the intersection of a right circular cone and a plane parallel to an element of the cone or by the locus of points equidistant from a fixed line and a fixed point not on the line.

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#### Hyperbola

A plane curve having two branches, formed by the intersection of a plane with both halves of a right circular cone at an angle parallel to the axis of the cone. It is the locus of points for which the difference of the distances from two given points is a constant.

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#### Parabola

(geometry) The conic section formed by the intersection of a cone with a plane parallel to a tangent plane to the cone; the locus of points equidistant from a fixed point (the focus) and line (the directrix). Category:en:Functions Category:en:Curves

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Jan 14, 2024

#### Hyperbola

(geometry) A conic section formed by the intersection of a cone with a plane that intersects the base of the cone and is not tangent to the cone. The function y(x) = 1/x draws a hyperbola. Category:en:Curves Category:en:Functions

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#### Parabola

(rhetoric) The explicit drawing of a parallel between two essentially dissimilar things, especially with a moral or didactic purpose. A parable.

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#### Hyperbola

A curve formed by a section of a cone, when the cutting plane makes a greater angle with the base than the side of the cone makes. It is a plane curve such that the difference of the distances from any point of it to two fixed points, called foci, is equal to a given distance. See Focus. If the cutting plane be produced so as to cut the opposite cone, another curve will be formed, which is also an hyperbola. Both curves are regarded as branches of the same hyperbola. See Illust. of Conic section, and Focus.

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Jan 14, 2024

#### Parabola

A kind of curve; one of the conic sections formed by the intersection of the surface of a cone with a plane parallel to one of its sides. It is a curve, any point of which is equally distant from a fixed point, called the focus, and a fixed straight line, called the directrix. See Focus.

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Jan 14, 2024

#### Hyperbola

An open curve formed by a plane that cuts the base of a right circular cone

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#### Parabola

A plane curve formed by the intersection of a right circular cone and a plane parallel to an element of the curve

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## Repeatedly Asked Queries

#### What is the focus of a parabola?

The focus of a parabola is a fixed point from which distances to any point on the curve are measured.

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Feb 12, 2024

#### What is a parabola in simple terms?

A parabola is a U-shaped curve that is symmetrical and can be seen in graphs of quadratic equations.

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#### Can a parabola and hyperbola have the same equation?

No, they have different standard equations reflecting their unique geometries.

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#### What are real-world examples of parabolas?

Parabolas are seen in satellite dishes, car headlights, and the paths of projectiles.

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Feb 12, 2024

#### Are hyperbolas used in architecture?

Yes, hyperbolas have been used in architectural designs for their aesthetic and structural properties.

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#### Are hyperbolas found in nature?

Yes, hyperbolas can describe certain celestial orbits and are used in navigation systems.

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#### What role do parabolas play in optics?

In optics, parabolas are used to focus light rays to a single point, as in telescopes and searchlights.

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Feb 12, 2024

#### How is the directrix related to a parabola?

The directrix is a fixed straight line, and every point on the parabola is equidistant from the focus and the directrix.

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Feb 12, 2024

#### How do the foci of a hyperbola compare to that of a parabola?

A hyperbola has two foci, and the distance between any point on the hyperbola and these foci is constant. A parabola has a single focus, equidistant from any point on the curve and the directrix.

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#### Is the hyperbola related to hyperbolic geometry?

Yes, hyperbolas are central in hyperbolic geometry, a non-Euclidean geometric space.

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#### Can hyperbolas represent real-life phenomena?

Yes, hyperbolas can model various real-life phenomena, such as the paths of celestial bodies and signal ranges in navigation systems.

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Feb 12, 2024

#### What are the asymptotes of a hyperbola?

Asymptotes are straight lines that the branches of a hyperbola approach but never touch.

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#### Can hyperbolas have horizontal and vertical orientations?

Yes, hyperbolas can be oriented both horizontally and vertically, depending on their equation.

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#### How does gravity relate to parabolas?

In a uniform gravitational field, without air resistance, projectiles follow a parabolic trajectory.

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#### What is the vertex of a parabola?

The vertex is the highest or lowest point on a parabola, where it changes direction.

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#### Can a parabola be rotated?

Yes, a parabola can be rotated, changing its orientation but not its fundamental properties.

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#### What is the importance of the axis of symmetry in a parabola?

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves.

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Feb 12, 2024

#### What is the difference in the formulas for parabolas and hyperbolas?

Parabolas typically have a squared term equal to a linear term, while hyperbolas have a difference of squared terms.

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Feb 12, 2024

#### Are the branches of a hyperbola symmetrical?

Yes, the two branches of a hyperbola are mirror images of each other across the asymptotes.

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Feb 12, 2024

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About Author

Written by

Shumaila SaeedShumaila Saeed, an expert content creator with 6 years of experience, specializes in distilling complex topics into easily digestible comparisons, shining a light on the nuances that both inform and educate readers with clarity and accuracy.