# Difference Between Parabola and Hyperbola

A conic section is a curve obtained when a plane intersects a cone at some specific angle. There are three conic sections –  ellipse, parabola, and hyperbola.

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

## Key Takeaways

1. Parabolas are U-shaped curves representing quadratic functions, with one axis of symmetry and a single vertex.
2. Hyperbolas consists of two distinct curves, representing points with a constant difference between distances between two foci.
3. Both parabolas and hyperbolas are conic sections, but they exhibit different shapes and mathematical properties, with parabolas having a single branch and hyperbolas having two branches.

## Parabola vs Hyperbola

Parabola is a U-shaped curve that is symmetrical around its axis. In contrast, a hyperbola is a type of curve that has two branches that open up or down and are symmetrical around their centre point. In math, they are represented by different equations and have different properties.

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A parabola is a single open curve that extends to 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 unit.

## What is Parabola?

A parabola is the locus of all the points 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 parallel direction (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 with a constant difference between 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 more significant 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 :

1. A parabola is a locus of all the points with an equal distance from a focus and a directrix. On the other hand, a hyperbola is a locus of all the issues for which the difference in distance between two foci is constant.
2. A parabola is an open curve with one focus and directrix, whereas a hyperbola is an open curve with two branches with two foci and directrices.
3. The eccentricity of a parabola is one, whereas the eccentricity of a hyperbola is more significant than one.
4. 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.
5. The equation for a parabola is y = ax². On the other hand, the equation for a hyperbola is x²/a² – y²/b² = 1.

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