Parametric Equations

# Parametric Equations

## Parametric Equations

Parametric equations are a pair of equations that expresses a group of quantities as explicit functions of a number of independent variables. The independent variables are called ‘parameters’ and denoted by ‘t’.

Parametric Equations of a Circle

The equation of a circle with radius a is written as x2 + y2 = a2 in Cartesian coordinates.

The equation of circle can be written in the parametric form as follows:

x = a cos t

y = a sin t

Note that the parametric equations are non-unique, so, the same quantities can be written as a number of different parameters.

A single parameter is usually represented by the parameter t.

Parametric Equations of a Parabola

The equation of a parabola in Cartesian form is y = x2 or x = y2.

The equation of parabola in parametric form can be written as follows:

x = t

y = t2

This conversion of equation from Cartesian form to parametric form is called parameterization.

It provides great efficiency when we integrate or differentiate the curves.

Parametric Equations of an Ellipse

The Cartesian equation of an ellipse is x2/a2 + y2/b2 = 1.

The equation of ellipse in parametric form can be written as follows:

x = a cos t

y = b sin t

where x and y are the coordinates of any point on the ellipse, and a and b are the radius on the x-axis and y-axis, respectively.

Parametric Equations of a Hyperbola

The Cartesian equation of a hyperbola is x2/a2 - y2/b2 = 1.

The equation of hyperbola in parametric form can be written as follows:

x = a sec t

y = b tan t

where x and y are the coordinates of any point on the hyperbola, and t is the parameter.

Parametric Equations of a Line

The Cartesian equation of a line is ax + by = c.

The equation of a line is typically written as y = mx + c

Where m is the slope of the line and c is the y-intercept of the line.

The equation of the line in parametric form can be written as follows:

x = x0 + at

y = y0 + bt

where (x0, y0) are the coordinates of any point on the line, and t is the parameter.

Related Topics:

Linear Equations in One Variable

Simultaneous Linear Equations