A conic section is a curve obtained when a plane intersects a cone at some specific angle. There are three types of conic sections – ellipse, parabola, and hyperbola.
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An ellipse is a planar curve that has two focal points, and somewhat resembles a circle. However, the parabola and hyperbola are confusing sections.
Parabola vs Hyperbola
The difference between a parabola and a hyperbola is that the parabola is a single open curve with eccentricity one, whereas a hyperbola has two curves with an eccentricity greater than one.
A parabola is a single open curve that extends till 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 branch.
|Parameter Of Comparison||Parabola||Hyperbola|
|Definition||A parabola is a locus of the points that have equal distance from a focus and a directrix.||A hyperbola is a locus of the points that have a constant difference from two foci.|
|Shape||The parabola is an open curve that has one focus and one directrix.||The hyperbola is an open curve with two branches that has two foci and two directrices.|
|Eccentricity||The non-negative eccentricity of a parabola is one.||The non-negative eccentricity e of a hyperbola is greater than one.|
|Intersection of Plane||The intersection of the plane is parallel (ideal case) to the slant height of the cone.||The intersection of the plane is parallel (ideal case) to the perpendicular height of the double cone.|
|General Equation||The general equation of the parabola is y = ax² , a ≠ 0||The general equation of the hyperbola is x²/a² – y²/b² = 1|
What is Parabola?
A parabola is the locus of all the points that are 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 direction parallel (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 that have a constant difference from 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 greater 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 :
- A parabola is a locus of all the points that have equal distance from a focus and a directrix. On the other hand, a hyperbola is a locus of all the points for which the difference in distance between two foci is constant.
- A parabola is an open curve having one focus and directrix, whereas a hyperbola is an open curve with two branches having two foci and directrices.
- The eccentricity of a parabola is one, whereas the eccentricity of a hyperbola is greater than one.
- 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.
- 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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