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Lesson: Parametric Equations of Plane Curves

Worksheet • 6 Questions

Q1:

Use the fact that c o s h s i n h 2 2 π‘₯ βˆ’ π‘₯ = 1 to find a parametrization of the part of the hyperbola π‘₯ 2 5 βˆ’ 𝑦 8 1 = 1 2 2 that contains the point ( βˆ’ 5 , 0 ) .

  • A π‘₯ = βˆ’ 5 𝑑 , 𝑦 = 9 𝑑 c o s h s i n h
  • B π‘₯ = 2 5 𝑑 , 𝑦 = 8 1 𝑑 c o s h s i n h
  • C π‘₯ = 5 𝑑 , 𝑦 = 9 𝑑 c o s h s i n h
  • D π‘₯ = βˆ’ 5 𝑑 , 𝑦 = βˆ’ 9 𝑑 c o s h s i n h
  • E π‘₯ = 5 𝑑 , 𝑦 = βˆ’ 9 𝑑 c o s h s i n h

Q2:

A particle following the parameterization , of the unit circle starts at and moves counterclockwise. At what values of is the particle at ? Give exact values.

  • A
  • B
  • C
  • D
  • E

Q3:

The diagram shows a parabola that is symmetrical about the π‘₯ -axis and whose vertex is at the origin. It can be described by the parametric equations π‘₯ = 𝑝 𝑑 2 and 𝑦 = 2 𝑝 𝑑 , 𝑑 ∈ ℝ , where 𝑝 is a positive constant. The focus of the parabola is the point ( 𝑝 , 0 ) , and the directrix is the line with the equation π‘₯ + 𝑝 = 0 .

Find a pair of parametric equations that describe the parabola whose focus is the point ο€Ό 3 2 , 0  and whose directrix is the line π‘₯ = βˆ’ 3 2 . Include the parameter range.

  • A π‘₯ = 3 2 𝑑 2 , 𝑦 = 3 𝑑 , 𝑑 ∈ ℝ
  • B π‘₯ = ο€Ό 3 2 𝑑  2 , 𝑦 = 3 𝑑 , 𝑑 ∈ ℝ
  • C π‘₯ = 3 2 𝑑 2 , 𝑦 = 3 𝑑 , 𝑑 β‰₯ 0
  • D π‘₯ = 3 𝑑 , 𝑦 = 3 2 𝑑 2 , 𝑑 β‰₯ 0
  • E π‘₯ = 3 𝑑 , 𝑦 = 3 2 𝑑 2 , 𝑑 ∈ ℝ
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