A Simple Note on Parametric Form of a Hyperbola

The parabola is the curve describing the flight of a projectile in a vacuum and has found applications as an efficient source of solar energy, among other applications. It is also one of the four types of conic sections. Parametric equations are a compact form for writing down equations that must be used repeatedly. Parametric equations are especially useful for describing curves. This article will use the parabola as an example parametric form for hyperbola. This article will also give you a formula that can be used to calculate the parametric equation of hyperbola.

What is a Parabola?  

A parabola is a graph of a quadratic equation in Cartesian coordinates that opens to the right if it’s plotted vertically and to the left if it’s plotted horizontally. The graph of this equation looks like a “U” if it’s plotted in perspective. To draw a parabola in Cartesian coordinates, select the y-axis and the x-axis. Drag the quadratic equation into the Cartesian coordinates.

The Parametric Equation of a Parabola

The Parametric equation of a parabola can be written in the following parametric form:

x = -y2 + K (t)

y = x2 + K (t)

Where x and y are the abscissas, a and b are the endpoints of the interval on which the parabola is described, K(t) is a constant (known as a parameter), t is an independent variable. This parametric equation describes a closed curve in the form of an ellipse.

What is a Hyperbola?

The name of hyperbola comes from its shape since it is similar to an “S.” A hyperbola is the graph of second-degree equations that opens to the left if it’s plotted vertically and to the right, if it’s plotted horizontally. The cross-section of a hyperbola is always a conic section. A hyperbola has two foci. It means that any point on one side of the diameter is focused on each of the foci and any point on the diameter.

Parametric Equation of Hyperbola

A hyperbola is a type of conic section. It is the set of all points in space that have the same distance from two fixed points, called foci. The two foci are denoted by F 1 and F 2 . The points are usually referred to with numbers, starting at 0, with one focus as (0,0), and the other as simply (a). The line joining the two foci is called the directrix, and it passes through the two foci in a straight line and is always parallel to them.

A hyperbola with a center at (x=0, y=0 ) and distance 2a from the x-axis and y-axis intersects the x-axis at point P.

The hyperbola is called symmetrical if its asymptotes are also symmetrical.

Parametric Form of Hyperbola

In a Cartesian coordinate system, we have the parametric equation of a parabola given by

(x-h)2/a2 + (y-k)2/b2 = 1

Where (h, k) is the vertex and a, b are related to the length of its latus rectum. If a and b are equal to 1 or -1, the parabola becomes a horizontal or vertical line at y = x.

Hyperbola has two asymptotes at x = a and x = b. The expression gives its latus rectum:

sqrt(a2-x2), and the conic constant C is given by

a2/(4*b) If a>1

then the parabola opens to the right, but if a<1, it opens to the left.

The parametric form of a hyperbola is given by:

(x-h)2/a2 + (y-k)2/b2 = 1, where h, k are related to the vertex and a, b are related to the length of its latus rectum.

Parabola equation in Cartesian coordinates:

y =  x2 – 4xh. √{(x-h)2/a2 + (y-k)2/b2} 

= |1 h k |

=| a b |

This gives us the parametric form of a hyperbola.

To find the parametric form of hyperbola, we solve the equation by substituting x and y with their values in Cartesian coordinates, as presented above.

a = 1, b = 2, h = -1, k = -2:

(x-h)2/a2 + (y-k)2/b2 = 1

= (x-1)2/12 + (y-k)2/22

= x4 – 4*x*h + √(x*h*22 + y*k)

= x4 – 4xh – 4*. Its parametric equation is:

x4 + 4h*√(x2 – x*h) + 4k = 1

The point where the curve crosses the coordinate axis is (-1, -2).

Conclusion

Hyperbola is the mathematical curve consisting of two mutually perpendicular branches to each other and separate. Hyperbola is also equivalent to a pair of lines with a common point, separated by an infinite distance. One branch is called the transverse axis, and the other is the direct axis. Parametric equations are a compact form for writing down equations that must be used repeatedly. They are especially useful for describing curves. The parametric expression for a hyperbola is a pair of equations that represent the direct and transverse axes simultaneously by applying specific values of the parameter