# Parametric Equations

## Summary

• Parametric equations are just rectangular equations consisting of two or more variables.
• At times it is convenient to express x and y in terms of a third variable which is called a parameter.
• Parametric equation includes one equation to define each variable.
• For example in parametric equations: x = a cos(t) and y = a sin(t), t is known as the parameter or third variable.
• To evaluate parametric equations, we need to eliminate the parameter from the equations using different identities, rules or formulas.

#### What are parametric equations?

Parametric equations are just rectangular equations consisting of two or more variables and defines each variable in terms of one parameters. Parametric equation includes one equation to define each variable ie an equation like x + y + z = a includes 3 variables x, y and z hence this equation will have three parametric equations, one for each variable.

Furthermore, we say that a relation between the coordinates x and y is known as ”cartesian form”. However, there are times when it is convenient to express x and y in terms of a third variable which is called a parameter. A parameter is a variable used in two equations that completely describe a function.

Suppose, the equation of a curve is given by two parametric equations. For example, two parametric equations of a circle with centre zero and radius a are given by:

x = a cos(t) and y = a sin(t) here t is the parameter.

To find the cartesian form, we must eliminate the third variable t from the above two equations as we only need an equation y in terms of x. For different parametric equations, different methods, rules and identities can be used. In this case we know that to eliminate variable t, we square both equations and add them ie:

${ x }^{ 2 }\quad +\quad { y }^{ 2 }\quad =\quad { a }^{ 2 }{ cos }^{ 2 }(t)\quad +\quad { a }^{ 2 }{ sin }^{ 2 }(t)$

We know the identity  ${ sin }^{ 2 }(\theta ){ \quad +\quad cos }^{ 2 }(\theta )\quad =\quad 1$

Hence:

${ x }^{ 2 }\quad +\quad { y }^{ 2 }\quad =\quad { a }^{ 2 }\left[ { cos }^{ 2 }(t)\quad +\quad { sin }^{ 2 }(t) \right]$

${ x }^{ 2 }\quad +\quad { y }^{ 2 }\quad =\quad { a }^{ 2 }(1)$

$\quad { y }^{ 2 }\quad =\quad { a }^{ 2 }\quad -\quad { x }^{ 2 }$

We will now go through a few examples to understand how parametric equations are evaluated.

#### Example #1

Q. Find the cartesian form from the given parametric equations.

$x\quad =\quad 4\quad +\quad { 4t }^{ 2 }$,

$y\quad =\quad { t }^{ 2 }\quad -\quad 3t$

Solution:

Say:

$x\quad =\quad 4\quad +\quad 3{ t }^{ 2 }\quad \quad \quad \Rightarrow \quad \quad (i)$

$y\quad =\quad 1\quad -\quad 3t\quad \quad \quad \Rightarrow \quad \quad (ii)$

Make t the subject of any one of the above equations. We choose equation (ii):

$3t\quad =\quad 1\quad -\quad y$

$t\quad =\quad \frac { 1\quad -\quad y }{ 3 }$

Put t in equation (i):

$x\quad =\quad 4\quad +\quad 3(\frac { 1\quad -\quad y }{ 3 } )$

$x\quad =\quad 4\quad +\quad 1\quad -\quad y$

$y\quad =\quad 5\quad -\quad x$      Ans

#### Example #2

Q. Find the equation of a circle from the given parametric equations:

x = 3 cos(t) y = 3 sin(t)

Solution:

As discussed previously, we square and add both the equations:

${ x }^{ 2 }{ \quad =\quad 3 }^{ 2 }{ cos }^{ 2 }(t)$

${ y }^{ 2 }{ \quad =\quad 3 }^{ 2 }{ sin }^{ 2 }(t)$

${ x }^{ 2 }\quad +\quad { y }^{ 2 }{ \quad =\quad { 9cos }^{ 2 }(t)\quad +\quad }9{ sin }^{ 2 }(t)$

Using the identity:   ${ sin }^{ 2 }(\theta ){ \quad +\quad cos }^{ 2 }(\theta )\quad =\quad 1$

${ x }^{ 2 }\quad +\quad y^{ 2 }\quad =\quad 9\left[ { cos }^{ 2 }(t){ \quad +\quad sin }^{ 2 }(t) \right]$

${ x }^{ 2 }\quad +\quad y^{ 2 }\quad =\quad 9$    This is the equation of a circle

#### Example #3

Q. What is the Cartesian equation given parametrically by equations:

$x\quad =\quad 2sin(\theta )$,

$y\quad =\quad 2sin(2\theta )$

Solution:

We know the identity of a double angle formula:

$sin(2\theta )\quad =\quad 2sin(\theta )\quad cos(\theta )$

Using this identity make  $\theta$ the subject of equation in the second parametric equation:

$y\quad =\quad 2\left[ 2sin(\theta )cos(\theta ) \right]$

$y\quad =\quad 4sin(\theta )cos(\theta )\quad \quad \quad \Rightarrow \quad \quad \quad (i)$

$\frac { x }{ 2 } \quad =\quad sin(\theta )\quad \quad \quad \Rightarrow \quad \quad \quad (ii)$

Square equation (i):

${ y }^{ 2 }\quad =\quad 16{ sin }^{ 2 }(\theta ){ cos }^{ 2 }(\theta )\quad \quad \quad \Rightarrow \quad \quad (iii)$

We know another identity:   ${ cos }^{ 2 }(\theta )\quad =\quad 1\quad -\quad { sin }^{ 2 }(\theta )$

We know:   $sin(\theta )\quad =\quad \frac { x }{ 2 }$

${ cos }^{ 2 }(\theta )\quad =\quad 1\quad -\quad { (\frac { x }{ 2 } ) }^{ 2 }$

${ cos }^{ 2 }(\theta )\quad =\quad 1\quad -\quad \frac { { x }^{ 2 } }{ 4 }$

Put value of  ${ cos }^{ 2 }(\theta )$  from above in equation (iii):

$y^{ 2 }\quad=\quad 16{ sin }^{ 2 }(\theta )(1\quad -\quad \frac { { x }^{ 2 } }{ 4 } )$

Now put the value of  ${ sin }^{ 2 }(\theta )$  in the above equation as well:

$y^{ 2 }\quad =\quad 16(\frac { { x }^{ 2 } }{ 4 } )(1\quad -\quad \frac { { x }^{ 2 } }{ 4 } )$

Lastly, simplifying the equation gives us:

$y^{ 2 }\quad =\quad 4{ x }^{ 2 }(1\quad -\quad \frac { { x }^{ 2 } }{ 4 } )$       Ans

#### Example #4

Q. Find the cartesian form from the given parametric equations:

$x\quad =\quad 4t\quad +\quad 2\quad \quad \quad \Rightarrow \quad \quad (i)$,

$y\quad =\quad { t }^{ 2 }\quad \quad \quad \Rightarrow \quad \quad (ii)$

Solution:

Firstly make t the subject of any one of the above equations. We choose equation (i):

$t\quad =\quad \frac { x\quad -\quad 2 }{ 4 } \quad \quad \rightarrow \quad \quad \frac { x }{ 4 } \quad -\quad \frac { 1 }{ 4 }$

Now put value of t in equation (ii):

$y\quad =\quad { (\frac { x }{ 4 } \quad -\quad \frac { 1 }{ 4 } ) }^{ 2 }$

Simplifying gives us the equation:

$y\quad =\quad \frac { { x }^{ 2 } }{ 16 } \quad -\quad { \frac { x }{ 4 } \quad +\quad \frac { 1 }{ 4 } }$      Ans