Parametric equations are a cool way to describe curves using a parameter like time or angle. They're super useful for modeling real-world situations, from projectile motion to planetary orbits.

Converting between parametric and Cartesian forms is key. It helps identify conic sections and their features. Mastering these conversions opens up new ways to analyze and solve problems in geometry and physics.

Parametric Equations and Cartesian Form

Elimination of parametric equations

• Parametric equations express x and y coordinates in terms of a parameter t $x = f(t)$, $y = g(t)$ (time, angle)
• Elimination process:
  1. Solve one equation for t in terms of x or y
  2. Substitute t expression into other equation
  3. Simplify resulting equation
• Common techniques:
  • Direct substitution replaces t with equivalent expression
  • Algebraic manipulation rearranges equations
  • Trigonometric identities used for sine and cosine equations ($\sin^2t + \cos^2t = 1$)

Identification of conic sections

• Four types: circle, ellipse, parabola, hyperbola
• Analyze eliminated Cartesian equation:
  • Circle: $(x - h)^2 + (y - k)^2 = r^2$
  • Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$
  • Parabola: $(y - k)^2 = 4p(x - h)$ or $(x - h)^2 = 4p(y - k)$
  • Hyperbola: $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$
• Special cases include degenerate conics (point, line, pair of lines) and rotated conics

Equations from parametric form

• Eliminate parameter to obtain Cartesian equation
• Identify resulting equation type
• Determine key features:
  • Center (h, k)
  • Vertices define shape's extent
  • Foci influence curve's shape
  • Eccentricity measures deviation from circular form

Conversion to parametric form

• Reverse process of parameter elimination
• General strategies:
  • Use trigonometric functions for circles and ellipses
  • Represent one variable in terms of other using parameter t
• Specific techniques:
  • Circle: $x = r\cos(t)$, $y = r\sin(t)$
  • Ellipse: $x = a\cos(t)$, $y = b\sin(t)$
  • Parabola: $x = at^2$, $y = 2at$ (vertical) or $x = 2at$, $y = at^2$ (horizontal)
  • Hyperbola: $x = a\sec(t)$, $y = b\tan(t)$ or $x = a\cosh(t)$, $y = b\sinh(t)$
• Verify conversion by eliminating parameter to recover original Cartesian equation