## 17Calculus Parametric Equations - Eliminating The Parameter

##### 17Calculus

On this page we discuss how to eliminate the parameter from parametric equations.

### Resources

precalculus

basics of parametrics

Wikipedia - Parametric Equations

Eliminating The Parameter

Sometimes we are given the set of parametric equations and we are asked to write the equation without the parameter by eliminating the parameter. This not always possible but with some equations there are ways to do it. Here are some of the ways.

Solve For $$t$$ and Substitute

The easiest technique to try is to solve one of the equations for the parameter and then substitute the result in the other equation. Here is an example.
$$x = 2t$$ and $$y = t^2$$
Solve the first equation for $$t$$ giving $$t=x/2$$ and substitute into the second equation.
$$y = (x/2)^2 = x^2/4$$

WARNING - This technique can sometimes lead to errors if you are not careful. One example is shown below in practice problem 4423. You need to be careful to watch that you do not lose information about the domain and range when you perform algebra.

For example, if you have $$x=\sqrt{t}$$ and you square both sides to get $$x^2=t$$. Now, even though you have performed a valid operation, these equations are NOT the same. In the first one, $$x$$ and $$t$$ are both greater than or equal to zero. However, in the second one, $$x$$ can be negative. So certain operations require you to keep the domain (when working with $$x$$) and the range (when working with $$y$$) in mind.

Here are some situations you need to watch out for.
1. when you taking the even power or even root of both sides of an equation,
2. when you are working with trig functions and taking inverses
3. when you are working with logarithms
4. when you dividing by a term, make sure it is never zero

Use Trig Identities

A second technique involves the use of trig identities. For example, given the parametric equations $$x=\cos(t)$$ and $$y=\sin(t)$$, we know that $$\cos^2(x) + \sin^2(x) = 1$$. So we can write $$x^2 + y^2 = 1$$ which eliminates the parameter.

By Inspection

A third way is by inspection. Sometimes it is obvious what a substitution might be. For example, if we have $$x=e^t$$ and $$y=e^{3t}+1$$ then we can rewrite $$y=(e^t)^3+1$$. Then we can replace $$e^t$$ with $$x$$ in the last equation (because our first parametric equation was $$x=e^t$$) to get $$y=x^3+1$$. Of course, the first technique would have worked too by solving $$x=e^t$$ for $$t$$ and then substituting and simplifying but by standing back and looking at the equations more carefully, the solution was much easier.
Okay, time for some practice problems.

Practice

Unless otherwise instructed, eliminate the parameter from these parametric equations. State restrictions on the domain and range, if any exist.

Eliminate the parameter $$t$$ from $$x=3t$$ and $$y=t^2$$.

Problem Statement

Eliminate the parameter $$t$$ from $$x=3t$$ and $$y=t^2$$.

$$y = x^2/9$$

Problem Statement

Eliminate the parameter $$t$$ from $$x=3t$$ and $$y=t^2$$.

Solution

Solve the first equation for t giving $$t=x/3$$ and substitute into the second equation.
$$y = (x/3)^2 = x^2/9$$

$$y = x^2/9$$

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$$x=\sqrt{t}$$, $$y=1-t$$

Problem Statement

Eliminate the parameter from the parametric equations $$x=\sqrt{t}$$, $$y=1-t$$

$$y = 1-x^2$$, $$x \geq 0$$

Problem Statement

Eliminate the parameter from the parametric equations $$x=\sqrt{t}$$, $$y=1-t$$

Solution

Here is one of the times you need to be careful about domain. In the video, he squares both sides of $$x=\sqrt{t}$$ but doesn't mention that doing so requires you to be careful about the domain. In this case, the correct answer requires you to say that $$x \geq 0$$

### PatrickJMT - 4423 video solution

video by PatrickJMT

$$y = 1-x^2$$, $$x \geq 0$$

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Eliminate the parameter $$t$$ from $$x=3\cos(t)$$ and $$y=2\sin(t)$$.

Problem Statement

Eliminate the parameter $$t$$ from $$x=3\cos(t)$$ and $$y=2\sin(t)$$.

$$\displaystyle{ \frac{x^2}{9} + \frac{y^2}{4} = 1 }$$

Problem Statement

Eliminate the parameter $$t$$ from $$x=3\cos(t)$$ and $$y=2\sin(t)$$.

Solution

We will use the trig identity $$\cos^2(t) + \sin^2(t) = 1$$.

$$x = 3\cos(t) \to x/3 = \cos(t)$$

$$y = 2\sin(t) \to y/2 = \sin(t)$$

$$(x/3)^2 + (y/2)^2 = 1$$

$$\displaystyle{ \frac{x^2}{9} + \frac{y^2}{4} = 1 }$$

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Eliminate the parameter t from $$x=5\cos(t)$$ and $$y=3\sin(t)$$.

Problem Statement

Eliminate the parameter t from $$x=5\cos(t)$$ and $$y=3\sin(t)$$.

$$9x^2 + 25y^2 = 225$$

Problem Statement

Eliminate the parameter t from $$x=5\cos(t)$$ and $$y=3\sin(t)$$.

Solution

In the video, she shows a really difficult way to solve this problem. Her solution is correct but it is overly tedious and unnecessarily complicated. Here is an easier and more common solution.
In this problem, you are given that $$x=5\cos(t)$$ and $$y=3\sin(t)$$. We solve each of these for the sine and cosine terms since we use $$\cos^2(t)+\sin^2(t)=1$$. So we have $$x/5=\cos(t) \to (x/5)^2=\cos^2(t)$$ and $$y/3=\sin(t) \to (y/3)^2=\sin^2(t)$$.
$$\begin{array}{rcl} 1 & = & \cos^2(t)+\sin^2(t) \\ 1 & = & (x/5)^2+(y/3)^2 \\ 1 & = & x^2/25+y^2/9 \\ 225 & = & 9x^2 + 25y^2 \end{array}$$

### Krista King Math - 719 video solution

video by Krista King Math

$$9x^2 + 25y^2 = 225$$

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Eliminate the parameter from the parametric equations $$x=t-1$$, $$y=4t+1$$.

Problem Statement

Eliminate the parameter from the parametric equations $$x=t-1$$, $$y=4t+1$$.

Solution

### PatrickJMT - 48 video solution

video by PatrickJMT

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Eliminate the parameter from the parametric equations $$x=\sqrt{t+1}$$, $$y=3t+2$$.

Problem Statement

Eliminate the parameter from the parametric equations $$x=\sqrt{t+1}$$, $$y=3t+2$$.

Solution

### PatrickJMT - 49 video solution

video by PatrickJMT

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Eliminate the parameter t from $$x=\ln t$$ and $$y=\sqrt{t}$$.

Problem Statement

Eliminate the parameter t from $$x=\ln t$$ and $$y=\sqrt{t}$$.

Solution

### Michel vanBiezen - 2466 video solution

video by Michel vanBiezen

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Eliminate the parameter $$t$$ from $$x=e^t-1$$ and $$y=e^{2t}$$.

Problem Statement

Eliminate the parameter $$t$$ from $$x=e^t-1$$ and $$y=e^{2t}$$.

$$y=x^2+2x+1$$

Problem Statement

Eliminate the parameter $$t$$ from $$x=e^t-1$$ and $$y=e^{2t}$$.

Solution

### Michel vanBiezen - 2464 video solution

video by Michel vanBiezen

$$y=x^2+2x+1$$

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Eliminate the parameter from the parametric equations $$x=4e^{t/4}$$, $$y=3e^t$$.

Problem Statement

Eliminate the parameter from the parametric equations $$x=4e^{t/4}$$, $$y=3e^t$$.

Solution

### PatrickJMT - 50 video solution

video by PatrickJMT

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