Worksheet: Conversion between Parametric and Rectangular Equations

In this worksheet, we will practice converting from the parametric form of an equation to its equivalent rectangular form and vice versa.

Q1:

Use the fact that coshsinh๏Šจ๏Šจ๐‘ฅโˆ’๐‘ฅ=1 to find a parametrization of the part of the hyperbola ๐‘ฅ25โˆ’๐‘ฆ81=1๏Šจ๏Šจ that contains the point (โˆ’5,0).

  • A๐‘ฅ=5๐‘ก,๐‘ฆ=9๐‘กcoshsinh
  • B๐‘ฅ=โˆ’5๐‘ก,๐‘ฆ=โˆ’9๐‘กcoshsinh
  • C๐‘ฅ=25๐‘ก,๐‘ฆ=81๐‘กcoshsinh
  • D๐‘ฅ=โˆ’5๐‘ก,๐‘ฆ=9๐‘กcoshsinh
  • E๐‘ฅ=5๐‘ก,๐‘ฆ=โˆ’9๐‘กcoshsinh

Q2:

The first figure shows the graphs of cos2๐œ‹๐‘ก and sin2๐œ‹๐‘ก that parameterize the unit circle for 0โ‰ค๐‘กโ‰ค1. What do the two functions graphed in the second figure parameterize?

  • AThe unit circle
  • BThe square on (0.5,0.5), (0.5,โˆ’0.5), (โˆ’0.5,โˆ’0.5), (โˆ’0.5,0.5)
  • CThe square on (0,0), (0,1), (1,1), (0,1)
  • DThe square on (1,0), (0,1), (โˆ’1,0), (0,โˆ’1)

Q3:

Consider the points ๐ด=(1,1) and ๐ต=(5,4).

What is the length of ๐ด๐ต?

Find a parameterization of the segment ๐ด๐ต over 0โ‰ค๐‘กโ‰ค1.

  • A๐‘ฅ=4(๐‘ก+1),๐‘ฆ=3(๐‘ก+1)
  • B๐‘ฅ=๐‘ก+4,๐‘ฆ=๐‘ก+3
  • C๐‘ฅ=4๐‘ก+1,๐‘ฆ=3๐‘ก+1
  • D๐‘ฅ=3๐‘ก+1,๐‘ฆ=4๐‘ก+1
  • E๐‘ฅ=3(๐‘ก+1),๐‘ฆ=4(๐‘ก+1)

Find ๐‘“ and ๐‘” so that ๐‘ฅ=๐‘“(๐‘ก), ๐‘ฆ=๐‘”(๐‘ก) parameterizes ๐ด๐ต over 0โ‰ค๐‘กโ‰ค5.

  • A๐‘“(๐‘ก)=4๐‘ก+1,๐‘”(๐‘ก)=3๐‘ก+1
  • B๐‘“(๐‘ก)=๐‘ก+4,๐‘”(๐‘ก)=๐‘ก+3
  • C๐‘“(๐‘ก)=3๐‘ก+1,๐‘”(๐‘ก)=4๐‘ก+1
  • D๐‘“(๐‘ก)=๐‘ก+45,๐‘”(๐‘ก)=๐‘ก+35
  • E๐‘“(๐‘ก)=45๐‘ก+1,๐‘”(๐‘ก)=35๐‘ก+1

Using the functions above for 0โ‰ค๐‘ โ‰ค5, what is the distance between the point (1,1) and the point (๐‘“(๐‘ ),๐‘”(๐‘ ))?

  • A๐‘ โˆ’1
  • B๐‘ +1
  • C๐‘ 
  • D๐‘ +2
  • E๐‘ โˆ’2

The parameterization of ๐ด๐ต above is an example of an {arc-length parameterization} of a plane curve. Find an arc-length parameterization ๐‘ฅ=๐‘“(๐‘ก), ๐‘ฆ=๐‘”(๐‘ก) of ๐ด๐ถ, where ๐ถ=(13,6) and the parameter starts at ๐‘ก=0. Give the interval used.

  • A๐‘ฅ=1213๐‘ก+1,๐‘ฆ=513๐‘ก+1 on 5โ‰ค๐‘กโ‰ค13
  • B๐‘ฅ=1213๐‘ก+1,๐‘ฆ=513๐‘ก+1 on 0โ‰ค๐‘กโ‰ค13
  • C๐‘ฅ=513๐‘ก+1,๐‘ฆ=1213๐‘ก+1 on 0โ‰ค๐‘กโ‰ค13
  • D๐‘ฅ=๐‘ก+1213,๐‘ฆ=513๐‘ก on 0โ‰ค๐‘กโ‰ค13
  • E๐‘ฅ=513๐‘ก+1,๐‘ฆ=1213๐‘ก+1 on 5โ‰ค๐‘กโ‰ค13

Q4:

A particle moves along the curve given by the parametric equations ๐‘ฅ=2๐‘ก+1๏Šจ, ๐‘ฆ=โˆ’3๐‘ก+2๏Šจ with โˆ’1โˆš2โ‰ค๐‘กโ‰ค1.

At which point is the particle located when ๐‘ก=1โˆš2? Give your answer to one decimal place, if necessary.

  • A(2,1)
  • B(2,0.5)
  • C(1.5,0.5)
  • D(1,2)
  • E(0.5,2)

At which points is the particle located when ๐‘ก=0 and ๐‘ก=1? Give your answers to one decimal place, if necessary.

  • A(2,โˆ’3) and (1,3)
  • B(1,2) and (3,1)
  • C(2,1) and (โˆ’1,3)
  • D(1,2) and (3,โˆ’1)
  • E(1,2) and (3,5)

Find an equation for the line that the particle moves along in the form ๐‘Ž๐‘ฅ+๐‘๐‘ฆ=๐‘.

  • A2๐‘ฅ+3๐‘ฆ=7
  • B3๐‘ฅ+2๐‘ฆ=7
  • C3๐‘ฅ+2๐‘ฆ=13
  • D3๐‘ฅโˆ’2๐‘ฆ=7
  • E3๐‘ฅโˆ’2๐‘ฆ=โˆ’1

What is the smallest value of ๐‘ฅ during the particleโ€™s motion? When is it reached?

  • A๐‘ฅ=1, at ๐‘ก=โˆ’1โˆš2
  • B๐‘ฅ=1, at ๐‘ก=0
  • C๐‘ฅ=0.5, at ๐‘ก=โˆ’1โˆš2
  • D๐‘ฅ=0.5, at ๐‘ก=0
  • E๐‘ฅ=โˆ’1, at ๐‘ก=1

Describe the motion from ๐‘ก=โˆ’1โˆš2 to ๐‘ก=1 in terms of position on the line.

  • AThe particle starts at (0.5,2), goes left and upward to (2,1), and then goes back right and downward to (โˆ’1,3).
  • BThe particle starts at (2,0.5), goes left and upward to (1,2), and then goes back right and downward to (3,โˆ’1).
  • CThe particle starts at (2,0.5), goes right and downward to (1,2), and then goes back left and upward to (3,โˆ’1).
  • DThe particle starts at (1,2), goes left and upward to (2,1), and then goes back right and downward to (3,โˆ’1).
  • EThe particle starts at (0.5,2), goes right and downward to (2,1), and then goes back left and upward to (โˆ’1,3).

Give the parameters ๐‘ฅ=๐‘“(๐‘ก), ๐‘ฆ=๐‘”(๐‘ก) that describe the same motion but on an interval starting at ๐‘ก=0 instead of โˆ’1โˆš2. In what interval does ๐‘ก lie?

  • A๐‘ฅ=๏€ฟ๐‘กโˆ’1โˆš2๏‹+1, ๐‘ฆ=๏€ฟ๐‘กโˆ’1โˆš2๏‹+1, the interval 0โ‰ค๐‘กโ‰คโˆš2+1โˆš2
  • B๐‘ฅ=2๏€ฟ๐‘ก+1โˆš2๏‹โˆ’1, ๐‘ฆ=โˆ’3๏€ฟ๐‘ก+1โˆš2๏‹โˆ’1, the interval 0โ‰ค๐‘กโ‰คโˆš2+1โˆš2
  • C๐‘ฅ=2๏€ฟ๐‘ก+1โˆš2๏‹+1, ๐‘ฆ=โˆ’3๏€ฟ๐‘ก+1โˆš2๏‹+1, the interval 0โ‰ค๐‘กโ‰คโˆš2+1โˆš2
  • D๐‘ฅ=๏€ฟ๐‘ก+1โˆš2๏‹+1, ๐‘ฆ=๏€ฟ๐‘ก+1โˆš2๏‹+1, the interval 0โ‰ค๐‘กโ‰คโˆš2+1โˆš2
  • E๐‘ฅ=2๏€ฟ๐‘กโˆ’1โˆš2๏‹+1๏Šจ, ๐‘ฆ=โˆ’3๏€ฟ๐‘กโˆ’1โˆš2๏‹+2๏Šจ, the interval 0โ‰ค๐‘กโ‰คโˆš2+1โˆš2

Q5:

Convert the parametric equations ๐‘ฅ=3๐‘กcos and ๐‘ฆ=3๐‘กsin to rectangular form.

  • A๐‘ฅโˆ’๐‘ฆ=9๏Šจ๏Šจ
  • B๐‘ฆ=3๐‘ฅsin
  • C๐‘ฅ+๐‘ฆ=3๏Šจ๏Šจ
  • D๐‘ฅ+๐‘ฆ=9๏Šจ๏Šจ
  • E๐‘ฅ=3๐‘ฆcos

Q6:

Convert the rectangular equation ๐‘ฅ+๐‘ฆ=25๏Šจ๏Šจ to parametric form.

  • A๐‘ฅ=๐‘ก,๐‘ฆ=25๐‘กsincos๏Šจ๏Šจ
  • B๐‘ฅ=5๐‘ก,๐‘ฆ=5๐‘กcossin
  • C๐‘ฅ=5๐‘ก,๐‘ฆ=5๐‘กcossin๏Šจ๏Šจ
  • D๐‘ฅ=๐‘ก,๐‘ฆ=๐‘กsincos
  • E๐‘ฅ=25๐‘ก,๐‘ฆ=25๐‘กsincos

Q7:

Convert the parametric equations ๐‘ฅ=๐‘ก+2๏Šจ and ๐‘ฆ=3๐‘กโˆ’1 to rectangular form.

  • A๐‘ฅ=๐‘ฆ+2๏Šจ
  • B๐‘ฅ=๐‘ฆ+13+2
  • C๐‘ฅ=๏€ผ๐‘ฆ+13๏ˆ+2๏Šจ
  • D๐‘ฅ=3๐‘ฆโˆ’1
  • E๐‘ฅ=๏€ผ๐‘ฆ+13๏ˆ๏Šจ

Q8:

Convert the rectangular equation ๐‘ฅโˆ’๐‘ฆ=9๏Šจ๏Šจ to parametric form.

  • A๐‘ฅ=3๐‘ก,๐‘ฆ=3๐‘กcoshsinh
  • B๐‘ฅ=9๐‘ก,๐‘ฆ=9๐‘กcoshsinh
  • C๐‘ฅ=3๐‘ก,๐‘ฆ=3๐‘กcossin
  • D๐‘ฅ=9๐‘ก,๐‘ฆ=9๐‘กsincos
  • E๐‘ฅ=3๐‘ก,๐‘ฆ=3๐‘กsinhcosh

Q9:

Consider the points ๐ด=(โˆ’1,1) and ๐ต=(4,2). Parameterize the segment ๐ด๐ต, where 0โ‰ค๐‘กโ‰ค1.

  • A๐‘ฅ=1โˆ’๐‘ก,๐‘ฆ=5๐‘กโˆ’1
  • B๐‘ฅ=๐‘ก+1,๐‘ฆ=5๐‘กโˆ’1
  • C๐‘ฅ=5๐‘ก+1,๐‘ฆ=1โˆ’๐‘ก
  • D๐‘ฅ=5๐‘ก+1,๐‘ฆ=1+๐‘ก
  • E๐‘ฅ=5๐‘กโˆ’1,๐‘ฆ=๐‘ก+1

Q10:

Convert the parametric equations ๐‘ฅ=2๐‘ก+1 and ๐‘ฆ=๐‘กโˆ’4 to the rectangular form.

  • A๐‘ฆ=๐‘ฅโˆ’92
  • B๐‘ฆ=2๐‘ฅ+1
  • C๐‘ฆ=๐‘ฅ+92
  • D๐‘ฆ=๐‘ฅโˆ’4
  • E๐‘ฆ=๐‘ฅโˆ’12

Q11:

Convert the rectangular equation (๐‘ฅ+3)+(๐‘ฆ+5)=9๏Šจ๏Šจ to parametric form.

  • A๐‘ฅ=3(๐‘ก)+3๐‘ฆ=3(๐‘ก)+5cosandsin
  • B๐‘ฅ=3(๐‘ก)โˆ’3๐‘ฆ=3(๐‘ก)โˆ’5cosandsin
  • C๐‘ฅ=3(๐‘ก)๐‘ฆ=3(๐‘ก)cosandsin
  • D๐‘ฅ=9(๐‘ก)+3๐‘ฆ=9(๐‘ก)+5cosandsin
  • E๐‘ฅ=9(๐‘ก)โˆ’3๐‘ฆ=9(๐‘ก)โˆ’5cosandsin

Q12:

Convert the parametric equations ๐‘ฅ=๏€ผ12๐‘ก๏ˆln and ๐‘ฆ=3๐‘ก๏Šจ to rectangular form.

  • A๐‘ฆ=12๐‘’๏Ž ๏Žก๏—
  • B๐‘ฆ=12๐‘’๏Šจ๏—
  • C๐‘ฆ=38๐‘’๏Šจ๏—
  • D๐‘ฆ=6๐‘’๏Šจ๏—
  • E๐‘ฆ=3๐‘’๏Šจ๏—

Q13:

Convert the parametric equations ๐‘ฅ=โˆš๐‘ก and ๐‘ฆ=5๐‘ก+4๐‘ก๏Šช to rectangular form.

  • A๐‘ฆ=5๐‘ฅ+4โˆš๐‘ฅ๏Šจ
  • B๐‘ฆ=5๐‘ฅ+5๐‘ฅ๏Šฎ๏Šจ
  • C๐‘ฆ=5๐‘ฅ+4๐‘ฅ๏Šช
  • D๐‘ฆ=5๐‘ฅโˆ’4๐‘ฅ๏Šฎ๏Šจ
  • E๐‘ฆ=5๐‘ฅ+4๐‘ฅ๏Šฎ๏Šจ

Q14:

Convert the parametric equations ๐‘ฅ=(๐‘ก)cos and ๐‘ฆ=(๐‘ก)sin to rectangular form.

  • A๐‘ฆ=(๐‘ฅ)sin
  • B๐‘ฅ+๐‘ฆ=1
  • C๐‘ฅ=(๐‘ฆ)cos
  • D๐‘ฅ+๐‘ฆ=1๏Šจ๏Šจ
  • E๐‘ฅโˆ’๐‘ฆ=1๏Šจ๏Šจ

Q15:

Convert the rectangular equation (๐‘ฅ+2)โˆ’(๐‘ฆ+5)=16๏Šจ๏Šจ to parametric form.

  • A๐‘ฅ=4(๐‘ก)โˆ’2๐‘ฆ=4(๐‘ก)โˆ’5coshandsinh
  • B๐‘ฅ=4(๐‘ก)๐‘ฆ=4(๐‘ก)coshandsinh
  • C๐‘ฅ=16(๐‘ก)โˆ’2๐‘ฆ=16(๐‘ก)โˆ’5coshandsinh
  • D๐‘ฅ=4(๐‘ก)โˆ’2๐‘ฆ=4(๐‘ก)โˆ’5cosandsin
  • E๐‘ฅ=4(๐‘ก)+2๐‘ฆ=4(๐‘ก)+5cosandsin

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