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Conics, Parametric Equations, and Polar Coordinates

10.2 Plane Curves and Parametric Equations

Focus on eliminate a parameter, orientation, coordinate evaluation.

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Ideas in this lesson

01Eliminate a parameterWorked steps · five questions02OrientationWorked steps · five questions03Coordinate evaluationWorked steps · five questions
Idea 1 of 3

Eliminate a parameter

Solve a suitable coordinate equation for the parameter and substitute, retaining the parameter restrictions.

Worked example 1

Let x(t)=2tx(t) = 2t and y(t)=t2y(t) = t^{2}. Find x(1)x(1).
  1. x=2t  ⟹  t=x2x = 2t \implies t = \dfrac{x}{2}
  2. y=t2=x24y = t^{2} = \dfrac{x^{2}}{4}
  3. x(1)=2(1)=2x(1) = 2(1) = 2

Answer: 22

Worked example 2

Let x(t)=3tx(t) = 3t and y(t)=t2y(t) = t^{2}. Find yy when x=6x = 6.
  1. x=3t  ⟹  t=x3x = 3t \implies t = \dfrac{x}{3}
  2. y=t2=x29y = t^{2} = \dfrac{x^{2}}{9}
  3. y(6)=22=4y(6) = 2^{2} = 4

Answer: 44

Idea 2 of 3

Orientation

The parameter determines traversal direction as well as the set of points. Check successive parameter values.

Worked example 1

For x=2cos⁡t,y=2sin⁡tx = 2 \cos t, y = 2 \sin t, how is the circle traced as tt increases from 00 to 2π2\pi ?
  1. t=0  ⟹  (x,y)=(2,0)t = 0 \implies (x,y) = (2,0)
  2. t=π2  ⟹  (x,y)=(0,2)t = \dfrac{\pi }{2} \implies (x,y) = (0,2)
  3. (2,0)→(0,2)  ⟹  (2,0) \to (0,2) \implies counterclockwise

Answer: Counterclockwise

Worked example 2

For x=3cos⁡t,y=3sin⁡tx = 3 \cos t, y = 3 \sin t, how is the circle traced as tt increases from 00 to 2π2\pi ?
  1. t=0  ⟹  (x,y)=(3,0)t = 0 \implies (x,y) = (3,0)
  2. t=π2  ⟹  (x,y)=(0,3)t = \dfrac{\pi }{2} \implies (x,y) = (0,3)
  3. (3,0)→(0,3)  ⟹  (3,0) \to (0,3) \implies counterclockwise

Answer: Counterclockwise

Idea 3 of 3

Coordinate evaluation

A single parameter value gives both coordinates of the same point. Do not use different parameter values for xx and yy.

Worked example 1

Let x(t)=2tx(t) = 2t and y(t)=t2y(t) = t^{2}. Find x(1)x(1).
  1. x=2t  ⟹  t=x2x = 2t \implies t = \dfrac{x}{2}
  2. y=t2=x24y = t^{2} = \dfrac{x^{2}}{4}
  3. x(1)=2(1)=2x(1) = 2(1) = 2

Answer: 22

Worked example 2

Let x(t)=3tx(t) = 3t and y(t)=t2y(t) = t^{2}. Find yy when x=6x = 6.
  1. x=3t  ⟹  t=x3x = 3t \implies t = \dfrac{x}{3}
  2. y=t2=x29y = t^{2} = \dfrac{x^{2}}{9}
  3. y(6)=22=4y(6) = 2^{2} = 4

Answer: 44

Resource alignment: Larson & Edwards, Calculus of a Single Variable, Tenth Edition, §10.2,p10.2, p. 696696. Explanations, diagrams and numerical practice variants are original MathLab content aligned to this section.