How to eliminate the parameter from parametric equations step-by-step

📚 Understanding Parametric Equations

Parametric equations define a set of quantities as functions of an independent variable, known as a parameter. Often, this parameter is denoted as $t$. Instead of directly relating $x$ and $y$, parametric equations express both $x$ and $y$ in terms of $t$.

📜 History and Background

The use of parameters to define curves and surfaces dates back to ancient Greek mathematics. However, the systematic study and application of parametric equations gained momentum with the development of analytic geometry in the 17th century. Mathematicians like Pierre de Fermat and René Descartes explored parametric representations of various curves.

🔑 Key Principles for Eliminating the Parameter

• 🔍 Solve for $t$ in one equation: Choose the simpler equation and isolate the parameter $t$.
• 💡 Substitute: Plug the expression for $t$ into the other equation.
• 📝 Simplify: Manipulate the resulting equation to eliminate $t$ and express $y$ as a function of $x$, or vice versa.
• 📈 Consider the Domain: Pay attention to any restrictions on $t$ which may affect the domain and range of the resulting Cartesian equation.

🧮 Step-by-Step Elimination Process

Here’s a detailed breakdown of how to eliminate the parameter $t$ from parametric equations:

1. Given Parametric Equations: You have two equations, $x = f(t)$ and $y = g(t)$.
2. Solve for $t$: Choose the simpler equation (either $x = f(t)$ or $y = g(t)$) and solve for $t$. For example, if $x = t + 2$, then $t = x - 2$.
3. Substitute: Substitute the expression you found for $t$ into the other equation. If $y = t^2$, substitute $t = x - 2$ to get $y = (x - 2)^2$.
4. Simplify: Simplify the resulting equation to obtain the Cartesian equation relating $x$ and $y$ directly. In our example, $y = (x - 2)^2$ is the Cartesian equation.

✏️ Example 1: Linear Equations

Let's consider the parametric equations $x = 2t + 1$ and $y = t - 3$.

• 🧪 Solve for $t$ in the second equation: $t = y + 3$.
• 🧬 Substitute into the first equation: $x = 2(y + 3) + 1$.
• 🔢 Simplify: $x = 2y + 6 + 1$, which gives $x = 2y + 7$. Rearranging, we get $y = \frac{1}{2}x - \frac{7}{2}$.

🌍 Example 2: Trigonometric Equations

Consider $x = 3\cos(t)$ and $y = 3\sin(t)$.

• 💡 Divide both equations: $\frac{x}{3} = \cos(t)$ and $\frac{y}{3} = \sin(t)$.
• 📝 Use the identity $\sin^2(t) + \cos^2(t) = 1$: $(\frac{x}{3})^2 + (\frac{y}{3})^2 = 1$.
• 📈 Simplify: $\frac{x^2}{9} + \frac{y^2}{9} = 1$, which means $x^2 + y^2 = 9$. This is a circle with radius 3.

📊 Example 3: More Complex Equations

Let $x = t^2$ and $y = 2t^3$.

• 🔑 Solve for $t$ using the first equation: $t = \sqrt{x}$.
• ✅ Substitute into the second equation: $y = 2(\sqrt{x})^3 = 2x^{3/2}$.

💡 Tips and Tricks

• 🧭 Choose the Right Equation: Select the equation that is easier to solve for $t$.
• 📐 Trigonometric Identities: When dealing with trigonometric functions, remember key identities like $\sin^2(t) + \cos^2(t) = 1$, $\tan(t) = \frac{\sin(t)}{\cos(t)}$, and others.
• 🧭 Domain Awareness: Always consider the domain of $t$ and how it affects the resulting Cartesian equation.

📝 Conclusion

Eliminating the parameter in parametric equations involves solving for the parameter in one equation and substituting it into the other. By following these steps and considering the domain, you can successfully convert parametric equations into Cartesian equations, making them easier to analyze and understand.