**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 x^{2} + y^{2} = a^{2}
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 = x^{2} or x = y^{2}.

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

x = t

y = t^{2}

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 x^{2}/a^{2} + y^{2}/b^{2}
= 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 x^{2}/a^{2} - y^{2}/b^{2}
= 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 = x_{0}
+ at

y = y_{0}
+ bt

where (x_{0},
y_{0}) are the coordinates of any point on the line, and t is the
parameter.

**Related Topics:**

**Linear Equations in One Variable**