The word parameter commonly means a feature or a characteristic that is useful in defining a system. In the disciplines like - logic, mathematics, languages and environmental sciences, the parameter has specific interpretations. In mathematics, the parameter is said to be a variable that represents a curve.

The parametric equations play a vital role in mathematics. These equations are the equations for a curve defining the coordinates of a point on the curve in the form of functions of a special variable which is known as parameter. These are the equations in set (that is more than one equation are there) that are used to express as explicit functions of various independent variables of a certain set of quantities which are commonly termed as parameters. In this article, we are going to learn about parametric equations and problems based on them.

Formula

In general, if we consider a two point system of x-y plane for a circle with equation $x^{2}+y^{2} = a^{2}$, then the parametric equations are given by:
x = a cos t
y = a sin t
Here, x and y are the general variables being now dependent on t the parameter. When these equations are taken together they serve as parametric equations.
It is to be noted that same quantity can be used to represent many parameterizations. Parameterization is an easy way for representing curves and other surfaces like circles, parabola, helix etc.

In general we can say that when we represent an equation of Cartesian form in x and y to a set of two equations by using a new parameter as x = g (t) and y = h (t), this is parameterization process and the new equations together are called parametric equations and the variable ‘t’ is more commonly termed as a parameter then.

The collection of points in the form (x, y) = (g (t), h (t)) can be plotted easily on the graph to obtain the curve representing the parametric equations.

Parametric Equations For Standard Curve

Let us have a look at parametric equations for some standard curves.

1) Circle:

Cartesian form : $x^{2} + y^{2} = a^{2}$

Parametric form: x = a cos t and y = a sin t

When a = 1 in above equation, the parametric form in rational format can be:

x = $\frac{1 - t^{2}}{1 + t^{2}}$ and y = $\frac{2t}{1 + t^{2}}$

2) Parabola:

Cartesian form: $y = x^{2}$

Parametric form: x = t, y = $t^{2}$, where t lies between $-\infty$ and $\infty$.

3) Ellipse:

Cartesian form: $\frac{x}{c^{2}}$ + $\frac{y}{d^{2}}$ = 1

Parametric form: x = c cos t and y = d sin t

Cartesian From Parametric

1) First obtain the value of parameter in either x or y whichever is easy using one equation. If we will be finding out the parameter’s value from the difficult equation or find out parameter in complex terms then calculation further would also get very much complex. So it is mandatory to look for the easiest equation to find the value of parameter.

2) Plug in this value in second equation and solve to get the required Cartesian form. It is not necessary to get a simple and reduced equation every time after this step. Also if any restriction on parameter levies then it should also be checked for the variables in Cartesian form accordingly.

Examples

Example 1: Obtain the parametric form for the equation below:

y = $\frac{3}{4} x^{2}$ + 4

Solution: Let x = 2u

This implies that y = $\frac{3}{4}$ $(2u)^{2}$ + 4 = 3$u^{2}$ + 4

Hence the parametric form for given equation is x = 2u and y = 3u$^{2}$ + 4.

Example 2: Eliminate t from x = $e^{3t}$ and y = $e^{t}$

Solution: x = $e^{3t}$ 

$\Rightarrow x = (e^{t})^{3}$

$\Rightarrow x = y^{3}$

This is the required Cartesian form.