# Eliminating Parameters in Parametric Curves

We are now going to look at taking a parametric curve $C$ defined by the parametric equations $x = f(t)$ and $y = g(t)$ and eliminate the parameter $t$ so that we write $C$ in terms of $x$ and $y$ only.

Note: While eliminating the parameter may be nice in verifying a type of curve, often times parameter eliminating will result in more complex equations. |

## Example 1

**Eliminate the parameter $t$ of the parametric curve $C$ defined by the equations $x = 3t + 1$ and $y = t^2$.**

We will first solve for $t$ by isolating it from the $x$-equation to get:

(1)Now we will substitute this value of $t$ into the $y$-equation as follows:

(2)We have thus eliminated the parameter $t$ and have written $C$ in terms of $x$ and $y$ only.

## Example 2

**Eliminate the parameter $t$ of the parametric curve $C$ defined by the equations $x = 3 \cos t$ and $y = 4 \sin t$.**

First let's isolate $t$ from the $x$-equation:

(3)Substituting this value of $t$ into the $y$-equation we get that:

(4)Now recall the following trigonometric identity $\sin ^2 a + \cos ^2 a = 1$. Suppose that we let $a = \cos ^{-1} x$, and then:

(5)It follows that:

(6)We have thus eliminated the parameter $t$.