Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.

Start Practicing

Worksheet: Parametric Equations of Plane Curves

Q1:

A particle following the parameterization π‘₯ = ( 2 πœ‹ 𝑑 ) c o s , 𝑦 = ( 2 πœ‹ 𝑑 ) s i n of the unit circle starts at ( 1 , 0 ) and moves counterclockwise. At what values of 0 ≀ 𝑑 ≀ 4 is the particle at ( 0 , 1 ) ? Give exact values.

  • A 1 4 , 3 4
  • B 1 4 , 5 4
  • C 1 4 , 3 4 , 5 4 , 7 4
  • D 1 4 , 5 4 , 9 4 , 1 3 4
  • E 1 4 , 3 4 , 5 4 , 7 4 , 9 4 , 1 1 4 , 1 3 4 , 1 5 4

Q2:

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

Q3:

The first figure shows the graphs of c o s 2 πœ‹ 𝑑 and s i n 2 πœ‹ 𝑑 , which parameterize the unit circle for 0 ≀ 𝑑 ≀ 1 . What do the two functions graphed in the second figure parameterize?

  • Athe unit circle
  • Bthe square on ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 1 ) , ( 0 , 1 )
  • Cthe square on ( 0 . 5 , 0 . 5 ) , ( 0 . 5 , βˆ’ 0 . 5 ) , ( βˆ’ 0 . 5 , βˆ’ 0 . 5 ) , ( βˆ’ 0 . 5 , 0 . 5 )
  • Dthe square on ( 1 , 0 ) , ( 0 , 1 ) , ( βˆ’ 1 , 0 ) , ( 0 , βˆ’ 1 )

Q4:

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

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

Q5:

A particle following the parameterization π‘₯ = ο€Ή 2 πœ‹ 𝑑  c o s  , 𝑦 = ο€Ή 2 πœ‹ 𝑑  s i n  of the unit circle starts at ( 1 , 0 ) and moves counterclockwise. At what values of 𝑑 β‰₯ 0 is the particle at ( 0 , 1 ) ? Give exact values.

  • A 𝑑 = ο„ž 1 4 + 𝑛 , where 𝑛 is a nonnegative integer
  • B 𝑑 = 1 4 + 𝑛 , where 𝑛 is a nonnegative integer
  • C 𝑑 = ο„ž 1 4 + 𝑛 , where 𝑛 is a nonnegative odd number
  • D 𝑑 = ο„ž 1 4 + 𝑛 , where 𝑛 is a nonnegative integer
  • E 𝑑 = ο„ž 1 4 + 2 𝑛 , where 𝑛 is a nonnegative integer

Q6:

Consider the points 𝐴 = ( 1 , 1 ) and 𝐡 = ( 5 , 4 ) .

What is the length of 𝐴 𝐡 ?

Find a parameterization of the segment 𝐴 𝐡 over 0 ≀ 𝑑 ≀ 1 .

  • A π‘₯ = 4 𝑑 + 1 , 𝑦 = 3 𝑑 + 1
  • B π‘₯ = 3 𝑑 + 1 , 𝑦 = 4 𝑑 + 1
  • C π‘₯ = 𝑑 + 4 , 𝑦 = 𝑑 + 3
  • D π‘₯ = 4 ( 𝑑 + 1 ) , 𝑦 = 3 ( 𝑑 + 1 )
  • E π‘₯ = 3 ( 𝑑 + 1 ) , 𝑦 = 4 ( 𝑑 + 1 )

Find 𝑓 and 𝑔 so that π‘₯ = 𝑓 ( 𝑑 ) , 𝑦 = 𝑔 ( 𝑑 ) parameterizes 𝐴 𝐡 over 0 ≀ 𝑑 ≀ 5 .

  • A 𝑓 ( 𝑑 ) = 𝑑 + 4 , 𝑔 ( 𝑑 ) = 𝑑 + 3
  • B 𝑓 ( 𝑑 ) = 4 𝑑 + 1 , 𝑔 ( 𝑑 ) = 3 𝑑 + 1
  • C 𝑓 ( 𝑑 ) = 4 5 𝑑 + 1 , 𝑔 ( 𝑑 ) = 3 5 𝑑 + 1
  • D 𝑓 ( 𝑑 ) = 𝑑 + 4 5 , 𝑔 ( 𝑑 ) = 𝑑 + 3 5
  • E 𝑓 ( 𝑑 ) = 3 𝑑 + 1 , 𝑔 ( 𝑑 ) = 4 𝑑 + 1

Using the functions above for 0 ≀ 𝑠 ≀ 5 , what is the distance between the point ( 1 , 1 ) and the point ( 𝑓 ( 𝑠 ) , 𝑔 ( 𝑠 ) ) ?

  • A 𝑠
  • B 𝑠 βˆ’ 2
  • C 𝑠 + 2
  • D 𝑠 + 1
  • E 𝑠 βˆ’ 1

The parameterization of 𝐴 𝐡 above is an example of an {arc-length parameterization} of a plane curve. Find an arc-length parameterization π‘₯ = 𝑓 ( 𝑑 ) , 𝑦 = 𝑔 ( 𝑑 ) of 𝐴 𝐢 , where 𝐢 = ( 1 3 , 6 ) and the parameter starts at 𝑑 = 0 . Give the interval used.

  • A π‘₯ = 1 2 1 3 𝑑 + 1 , 𝑦 = 5 1 3 𝑑 + 1 on 5 ≀ 𝑑 ≀ 1 3
  • B π‘₯ = 5 1 3 𝑑 + 1 , 𝑦 = 1 2 1 3 𝑑 + 1 on 0 ≀ 𝑑 ≀ 1 3
  • C π‘₯ = 𝑑 + 1 2 1 3 , 𝑦 = 5 1 3 𝑑 on 0 ≀ 𝑑 ≀ 1 3
  • D π‘₯ = 1 2 1 3 5 1 3 𝑑 + 1 t + 1 , y = on 0 ≀ 𝑑 ≀ 1 3
  • E π‘₯ = 5 1 3 𝑑 + 1 , 𝑦 = 1 2 1 3 𝑑 + 1 on 5 ≀ 𝑑 ≀ 1 3