Eliminating the Parameter: Techniques and Solved Examples

📚 Quick Study Guide

🔍 What is Eliminating the Parameter? It's the process of rewriting a set of parametric equations as a single equation in terms of $x$ and $y$, effectively removing the parameter (usually $t$ or $\theta$). 💡 Common Techniques:
• Substitution: Solve one equation for the parameter and substitute it into the other equation.
• Trigonometric Identities: Use identities like $\sin^2(\theta) + \cos^2(\theta) = 1$ when dealing with trigonometric parametric equations.
🔢 Algebraic Manipulation: Employ algebraic techniques like squaring, cubing, or taking reciprocals to eliminate the parameter. 📝 Key Formulas:
• If $x = f(t)$ and $y = g(t)$, solve for $t$ in one equation (e.g., $t = h(x)$) and substitute into the other: $y = g(h(x))$.
• For trigonometric parameters: Use identities like $\sin^2(t) + \cos^2(t) = 1$, $\sec^2(t) - \tan^2(t) = 1$, and $\csc^2(t) - \cot^2(t) = 1$.

Practice Quiz

1. Question 1: Given $x = t + 1$ and $y = t^2$, eliminate the parameter $t$.
1. $y = x^2 - 2x + 1$
2. $y = x^2 + 2x + 1$
3. $y = x^2 - 1$
4. $y = x^2 + 1$
2. Question 2: Given $x = 2\cos(\theta)$ and $y = 2\sin(\theta)$, eliminate the parameter $\theta$.
1. $x^2 + y^2 = 2$
2. $x^2 + y^2 = 4$
3. $x + y = 2$
4. $x - y = 2$
3. Question 3: Given $x = t^3$ and $y = t^2$, eliminate the parameter $t$.
1. $y = x^{\frac{2}{3}}$
2. $y = x^{\frac{3}{2}}$
3. $y = x^2$
4. $y = x^3$
4. Question 4: Given $x = e^t$ and $y = e^{2t}$, eliminate the parameter $t$.
1. $y = x$
2. $y = x^2$
3. $y = 2x$
4. $y = x^3$
5. Question 5: Given $x = \tan(\theta)$ and $y = \sec(\theta)$, eliminate the parameter $\theta$.
1. $y^2 - x^2 = 1$
2. $x^2 - y^2 = 1$
3. $x^2 + y^2 = 1$
4. $x + y = 1$
6. Question 6: Given $x = 3t - 1$ and $y = 6t + 2$, eliminate the parameter $t$.
1. $y = 2x + 4$
2. $y = 2x - 4$
3. $y = x + 4$
4. $y = x - 4$
7. Question 7: Given $x = \sqrt{t}$ and $y = t + 1$, eliminate the parameter $t$.
1. $y = x^2 + 1$
2. $y = x^2 - 1$
3. $y = x + 1$
4. $y = x - 1$