Hyperbola vs Rectangular Hyperbola
There are four types of conic sections called ellipse, circle, parabola and hyperbola. These four types of conic sections are formed by the intersection of a double-cone and a plane. Depending on the angle between the plane and the axis of the cone the type of the conic section will be decided. In this article, only the properties of hyperbola and the difference between hyperbola and rectangular hyperbola, which is a special case of hyperbola, are discussed.
The word “hyperbola” comes from a Greek word, which means “over-thrown”. It is believed that hyperbola was introduced by a great mathematician Apllonious.
There are two ways to form a hyperbola. First method is to consider the intersection between a cone and a plane, which is parallel to the axis of the cone. The second method is to consider the intersection between a cone and a plane, which makes an angle less than the angle between the axis of the cone and any line on the cone with the axis of the cone.
Geometrically hyperbola is a curve. The equation of the hyperbola can be written as (x2/a2) – (y2/b2) =1.
A hyperbola consists of two distinct branches, which are called connected components. The closest points on the two branches are called vertices and the line that pass through these two pints is called the major axis. As the two curves reach a larger distance from the center, they approach two lines. These lines are called asymptotes.
A special case of a hyperbola, in which a=b, in the equation of the hyperbola is called the rectangular hyperbola. Therefore, the equation of the rectangular hyperbola is x2 – y2 = a2.
The rectangular hyperbola has orthogonal asymptotic lines. The rectangular hyperbola is also called orthogonal hyperbola or equilateral hyperbola.
If the two curves of the rectangular parabola lie in the first and third quadrants of the coordinate plane with x-axis and y-axis, which is the asymptotes, then it is in the form of xy=k, where k is a positive number. If k is a negative number, the two branches of the rectangular hyperbola lay in the quadrants two and four.
What is the difference between ?
· Rectangular hyperbola is a special type of hyperbola in which it’s asymptotes are perpendicular to each other.
· (x2/a2) – (y2/b2) = 1 is the general form of hyperbolas, while a=b for rectangular hyperbolas, i.e: x2 – y2 = a2.