Parabola and hyperbola are two sections of a cone. You can understand these differences in a mathematical, conceptual design or understand them in a very simple way which is not only for mathematicians but very convenient for all to understand. The article below will try to explain these differences between Parabola and Hyperbola in a very simple way.
Where a cone is cut by a plane, the obtained section is called a conic section. Conic sections can possibly be circles, ellipses, hyperbolas, or parabolas depending upon the angle of intersection between the cone’s axis and the plane. Both the parabolas and the hyperbolas are open curves, which means that the arms or branches of the curves continue to be infinite. Parabola And hyperbola are not closed curves like a circle or an ellipse.
Parabola
A parabola is a curve formed when a plane cuts a cone parallel to the cone side. In the case of a parabola, the line passing through the focus and perpendicular to the directrix is known as an axis of symmetry. The point where the point intersects the parabola on the axis of symmetry is the vertex.
All the parabolas are shaped identically as they are cut at a particular angle. Hence, it is characterized by the eccentricity of 1. That’s the reason why they all are of the same shape but can possibly be of different sizes.
The equation of a parabola is given as y = ax²+bx+c, a ≠ 0.
When a set of points are present in a plane, equidistant from the directrix, a given straight line, and equidistant from the focus, and a specific point which is fixed, it is called a parabola.
Parabolas have various practical applications. For example, they are used for designing the path of missiles, radar receivers, headlight reflectors of automobiles, satellite dishes, telescopes, and much more.
Hyperbola
Hyperbola is the curve formed when the plane cuts almost parallel to the axis of a cone. Hyperbolas are not identical in shape. This is because of so many possible angles between the axis and the plane.
Vertices are the points on the two arms which are nearest. The line segment which joins these arms is known as the major axis. Unlike parabola, where the two arms of the curve, also called the branches, become parallel to each other, In a hyperbola, these two arms or curves never become parallel. The center of a hyperbola is the midpoint of the major axis.
The equation hyperbola is given as:
x²/a² – y²/b² = 1.
When the difference of distances between a set of points present in the plane to the two fixed foci or the points is a positive constant, it is called a hyperbola.
Difference between Parabola and Hyperbola
Definition
A parabola is a locus of the points equidistant from the focus and the directrix.
A hyperbola is a locus of the points equidistant from two foci.
Shape
The parabola is an open curve with only one focus and one directrix.
The hyperbola is an open curve with the two branches having two foci and two directrices.
Eccentricity
Eccentricity(e) of Parabola = 1.
Eccentricity(e) of Hyperbola> 1.
Intersection of Plane
The intersection of the plane is parallel in an ideal case to the slant height of the cone. The intersection of the plane is parallel in an ideal case to the perpendicular height of the double cone.
General Equation
The equation of the parabola is y = ax²+bx+c, a ≠ 0.
The equation of the hyperbola is x²/a² – y²/b² = 1.
Conclusion
Parabola and hyperbola are two sections of a cone. They are very different from each other, and these differences are discussed very profoundly in the article above. Hope this article helped you to get a better understanding of the topics Parabola and Hyperbola.
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