Introduction to hyperbola

HYPERBOLA

The plane which consists of a smooth curve defined by its geometric properties or by the equations of which it is a set of solutions is called a hyperbola. It is made of two parts namely arms, which are exact replicas of each other and similar to that of 2 bows that are infinite. A hyperbola is a conic section that has 2 branches and obtained by the intersection of a plane and a cone. The standard equation of the hyperbola is [(x – h)² / a²] – [(y – k)² / b²] = 1.

The below cases are a few instances where a hyperbola can be observed.

• The shape of an open orbit such as the orbit of a spaceship.
• The scattering trajectory of a subatomic particle.
• The path taken by the shadow of the tip of a sundial.
• The path traced by a single-apparition comet.

The parts of a hyperbola are explained below. 

• The two fixed points are called the foci.
• The hyperbola’s centre is the midpoint of the line segment joining the foci.
• The line which passes through the two fixed points (foci) is termed the transverse axis.
• The transverse axis is intersected by a few points called the vertices of the hyperbola.
• The line which passes through the centre and perpendicular to the transverse axis is the conjugate axis.
• A hyperbola is bounded by 2 imaginary lines called asymptotes.
• The chord length which is perpendicular to the major axis of the hyperbola and passing through one of the foci is the latus rectum.
• The semi latus rectum is half the length of the latus rectum.
• The eccentricity of a hyperbola is given by e = √1 + (b² / a²).
• The principal axes of hyperbola consist of the transverse axis and the conjugate axis.
• Focal distance is the distance from the foci to the centre. 

The above figure shows the different fragments that constitute a hyperbola. This article also answers the question – what are hyperbolas used for and also its applications. It includes:

• The airport of Dulles is in the shape of a hyperbolic paraboloid. A  hyperbolic paraboloid is the one which consists of a hyperbola in one part, and a parabola in another part.
• Any lampshade casts hyperbolic shadows on a wall.
• In gear transmission, the concept of two hyperboloids is used for revolution between the 2 skew axes.
• An alternate name for a hyperbola is “Sonic Boom Curve”. When the plane moves faster, a cone-wave is formed. When the cone is intersected by the ground, a hyperbola is obtained.
• The idea of a hyperbola is used in the design of cooling towers of nuclear reactors and coal-fired power plants. The two challenges that the engineers faced during the construction of the above two things are that the structure should be able to withstand high winds and they should be built with minimal material. These challenges were solved by the hyperbolic forms.
• Imagine a situation where 2 stones are thrown simultaneously in water resulting in ripples that are moving outward in concentric circles. A curve is obtained when these circles intersect at points called a hyperbola. 

Hyperbola finds its applications in most of the fields. A few of them are listed above for reference.