Worksheet: Parametric Equations of Plane Curves

In this worksheet, we will practice finding the parametric equations of a plane curve.

Q1:

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

  • A 𝑥 = 5 𝑡 , 𝑦 = 9 𝑡 c o s h s i n h
  • B 𝑥 = 5 𝑡 , 𝑦 = 9 𝑡 c o s h s i n h
  • C 𝑥 = 5 𝑡 , 𝑦 = 9 𝑡 c o s h s i n h
  • D 𝑥 = 5 𝑡 , 𝑦 = 9 𝑡 c o s h s i n h
  • E 𝑥 = 2 5 𝑡 , 𝑦 = 8 1 𝑡 c o s h s i n h

Q2:

A particle following the parameterization 𝑥 = ( 2 𝜋 𝑡 ) c o s , 𝑦 = ( 2 𝜋 𝑡 ) s i n of the unit circle starts at ( 1 , 0 ) and moves counterclockwise. At what values of 0 𝑡 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

Q3:

The diagram shows a parabola that is symmetrical about the 𝑥 -axis and whose vertex is at the origin. It can be described by the parametric equations 𝑥 = 𝑝 𝑡 2 and 𝑦 = 2 𝑝 𝑡 , 𝑡 , 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 𝑥 + 𝑝 = 0 .

Find a pair of parametric equations that describe the parabola whose focus is the point 3 2 , 0 and whose directrix is the line 𝑥 = 3 2 . Include the parameter range.

  • A 𝑥 = 3 𝑡 , 𝑦 = 3 2 𝑡 2 , 𝑡
  • B 𝑥 = 3 2 𝑡 2 , 𝑦 = 3 𝑡 , 𝑡 0
  • C 𝑥 = 3 𝑡 , 𝑦 = 3 2 𝑡 2 , 𝑡 0
  • D 𝑥 = 3 2 𝑡 2 , 𝑦 = 3 𝑡 , 𝑡
  • E 𝑥 = 3 2 𝑡 2 , 𝑦 = 3 𝑡 , 𝑡

Q4:

The first figure shows the graphs of c o s 2 𝜋 𝑡 and s i n 2 𝜋 𝑡 , which parameterize the unit circle for 0 𝑡 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 , 0 . 5 ) , ( 0 . 5 , 0 . 5 ) , ( 0 . 5 , 0 . 5 )
  • Dthe square on ( 1 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 0 , 1 )

Q5:

A particle following the parameterization 𝑥 = 2 𝜋 𝑡 c o s , 𝑦 = 2 𝜋 𝑡 s i n of the unit circle starts at ( 1 , 0 ) and moves counterclockwise. At what values of 𝑡 0 is the particle at ( 0 , 1 ) ? Give exact values.

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

Q6:

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

What is the length of 𝐴 𝐵 ?

Find a parameterization of the segment 𝐴 𝐵 over 0 𝑡 1 .

  • A 𝑥 = 4 𝑡 + 1 , 𝑦 = 3 𝑡 + 1
  • B 𝑥 = 3 𝑡 + 1 , 𝑦 = 4 𝑡 + 1
  • C 𝑥 = 𝑡 + 4 , 𝑦 = 𝑡 + 3
  • D 𝑥 = 4 ( 𝑡 + 1 ) , 𝑦 = 3 ( 𝑡 + 1 )
  • E 𝑥 = 3 ( 𝑡 + 1 ) , 𝑦 = 4 ( 𝑡 + 1 )

Find 𝑓 and 𝑔 so that 𝑥 = 𝑓 ( 𝑡 ) , 𝑦 = 𝑔 ( 𝑡 ) parameterizes 𝐴 𝐵 over 0 𝑡 5 .

  • A 𝑓 ( 𝑡 ) = 𝑡 + 4 , 𝑔 ( 𝑡 ) = 𝑡 + 3
  • B 𝑓 ( 𝑡 ) = 4 𝑡 + 1 , 𝑔 ( 𝑡 ) = 3 𝑡 + 1
  • C 𝑓 ( 𝑡 ) = 4 5 𝑡 + 1 , 𝑔 ( 𝑡 ) = 3 5 𝑡 + 1
  • D 𝑓 ( 𝑡 ) = 𝑡 + 4 5 , 𝑔 ( 𝑡 ) = 𝑡 + 3 5
  • E 𝑓 ( 𝑡 ) = 3 𝑡 + 1 , 𝑔 ( 𝑡 ) = 4 𝑡 + 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 2
  • C 𝑠 + 2
  • D 𝑠 + 1
  • E 𝑠 1

The parameterization of 𝐴 𝐵 above is an example of an {arc-length parameterization} of a plane curve. Find an arc-length parameterization 𝑥 = 𝑓 ( 𝑡 ) , 𝑦 = 𝑔 ( 𝑡 ) of 𝐴 𝐶 , where 𝐶 = ( 1 3 , 6 ) and the parameter starts at 𝑡 = 0 . Give the interval used.

  • A 𝑥 = 1 2 1 3 𝑡 + 1 , 𝑦 = 5 1 3 𝑡 + 1 on 5 𝑡 1 3
  • B 𝑥 = 5 1 3 𝑡 + 1 , 𝑦 = 1 2 1 3 𝑡 + 1 on 0 𝑡 1 3
  • C 𝑥 = 𝑡 + 1 2 1 3 , 𝑦 = 5 1 3 𝑡 on 0 𝑡 1 3
  • D 𝑥 = 1 2 1 3 𝑡 + 1 , 𝑦 = 5 1 3 𝑡 + 1 on 0 𝑡 1 3
  • E 𝑥 = 5 1 3 𝑡 + 1 , 𝑦 = 1 2 1 3 𝑡 + 1 on 5 𝑡 1 3

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