Introduction
Parametric equations like are powerful, but sometimes we want to see thefamiliar rectangular form . Eliminating the parameter means solving for in one equation and substituting into the other, revealing the underlying curve—whether it's a line, circle, parabola, or something else entirely.
Prerequisite Connection
You can graph parametric equations and understand how traces curves.
Today's Increment
We convert parametric equations to rectangular form .
Why This Matters
Rectangular form reveals what curve the parametric equations trace (line, circle, parabola, etc.).
Elimination Techniques
Method 1: Direct Substitution
Solve one equation for t, substitute into the other.
Method 2: Pythagorean Identity
For trig parametrics, use .
Method 3: Algebraic Manipulation
Add, subtract, or combine equations to eliminate t.
Worked Examples
Example 1: Substitution
Eliminate t:
Solve: , substitute into y.
Solution:
Example 2: Pythagorean Identity
Eliminate t:
Use
Solution: (ellipse)
Example 3: Algebraic
Eliminate t:
Subtract:
Solution: (for x ≥ 1)
Common Pitfalls
Losing domain restrictions - State any restrictions on x or y.
Wrong identity - Use Pythagorean only for sin/cos of same angle.
Real-World Application
Path Planning in Robotics
Robots receive parametric paths but need rectangular equations to avoid obstacles.
Practice Quiz
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