Parabola vs Hyperbola

Parabolas and Hyperbolas are geometric curves that form when a cone is sliced at different angles along a plane. Though they share a common origin in conic sections, they differ in a number of features, including their shape, eccentricity, and equation. 

Shape

A parabola is a symmetrical, U-shaped curve that represents the path of objects in projectile motion or the design of reflective surfaces like satellite dishes. In contrast, a hyperbola consists of two open, mirrored curves, or branches, that extend infinitely in opposite directions.

All parabolas have the same shape, regardless of their size, whereas hyperbolas can be of different shapes based on their eccentricity.

Mathematically, a parabola is defined as a set of points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix). A hyperbola is defined as the set of all points in a plane where the difference of distances to two fixed points (foci) is a positive constant.

The arms of a parabola are always parallel to each other, whereas the arms of a hyperbola are not. 

Thus, a parabola has one focus and one directrix. On the other hand, a hyperbola has two foci and two directrices.

Eccentricity (e)

The eccentricity of the parabola is always 1 (e = 1). In contrast, the eccentricity of the hyperbola is always greater than 1 (e > 1).

Equations

The hyperbolas can be of two forms – horizontal and vertical. On the other hand, parabolas can be in four forms along the x and y axes.

Parabola

The standard form of the equation of a parabola is

• y = ax² + bx + c
• x = ay² + by + c, a ≠ 0

The vertex form of the equation of a parabola is

• y = a(x – h)² + k
• x = a(y – k)² + h

Hyperbola

The standard form of the equation of a hyperbola is

• ${\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1}$
• ${\dfrac{y^{2}}{a^{2}}-\dfrac{x^{2}}{b^{2}}=1}$

The vertex form of the equation of a hyperbola is

• ${\dfrac{\left( x-h\right) ^{2}}{a^{2}}-\dfrac{\left( y-k\right) ^{2}}{b^{2}}=1}$
• ${\dfrac{\left( y-k\right) ^{2}}{a^{2}}-\dfrac{\left( x-h\right) ^{2}}{b^{2}}=1}$

Asymptotes

A hyperbola has two asymptotes that the curve approaches but never touches. In contrast, a parabola has no asymptotes.

Summary

Basis	Parabola	Hyperbola
Shape	A U-shaped curve that represents the path of objects in projectile motionHave the same shape regardless of their size	Two open, mirrored curves or branches that extend infinitely in opposite directionsHave different shapes based on their eccentricity
Foci and Directrix	Has one focus and one directrix	Has two branches, two foci, and two directrices
Eccentricity (e)	e = 1	e > 1
Arms	Parallel to each other	Parallel to each other
Asymptotes	Has no asymptotes	Has two asymptotes that the curve approaches but never touches
Forms	Can be of four forms: upward-facing, downward-facing, right-facing, and left-facing	Can be of two forms: horizontal and vertical
Equation	Standard Form: y = ax2 + bx + cx = ay2 + by + c, a ≠ 0Vertex Form:y = a(x – h)2 + kx = a(y – k)2 + h	Standard Form: ${\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1}$${\dfrac{y^{2}}{a^{2}}-\dfrac{x^{2}}{b^{2}}=1}$Vertex Form:${\dfrac{\left( x-h\right) ^{2}}{a^{2}}-\dfrac{\left( y-k\right) ^{2}}{b^{2}}=1}$${\dfrac{\left( y-k\right) ^{2}}{a^{2}}-\dfrac{\left( x-h\right) ^{2}}{b^{2}}=1}$