Worksheet: Conversion between Parametric and Rectangular Equations

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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 coshsinh𝑥−𝑥=1 to find a parametrization of the part of the hyperbola 𝑥25−𝑦81=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 = 2 5 𝑡 , 𝑦 = 8 1 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 = 5 𝑡 , 𝑦 = − 9 𝑡 c o s h s i n h

Q2:

The first figure shows the graphs of cos2𝜋𝑡 and sin2𝜋𝑡, 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 (0.5,0.5), (0.5,−0.5), (−0.5,−0.5), (−0.5,0.5)
  • Cthe square on (0,0), (0,1), (1,1), (0,1)
  • Dthe square on (1,0), (0,1), (−1,0), (0,−1)

Q3:

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

What is the length of 𝐴𝐵?

Find a parameterization of the segment 𝐴𝐵 over 0≤𝑡≤1.

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

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

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

The parameterization of 𝐴𝐵 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 𝐴𝐶, where 𝐶=(13,6) 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≤13
  • B 𝑥 = 1 2 1 3 𝑡 + 1 , 𝑦 = 5 1 3 𝑡 + 1 on 0≤𝑡≤13
  • C 𝑥 = 5 1 3 𝑡 + 1 , 𝑦 = 1 2 1 3 𝑡 + 1 on 0≤𝑡≤13
  • D 𝑥 = 𝑡 + 1 2 1 3 , 𝑦 = 5 1 3 𝑡 on 0≤𝑡≤13
  • E 𝑥 = 5 1 3 𝑡 + 1 , 𝑦 = 1 2 1 3 𝑡 + 1 on 5≤𝑡≤13

Q4:

A particle moves along the curve given by the parametric equations 𝑥=2𝑡+1, 𝑦=−3𝑡+2 with −1√2≤𝑡≤1.

At which point is the particle located when 𝑡=1√2? 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 𝑡=0 and 𝑡=1? Give your answers to one decimal place, if necessary.

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

Find an equation for the line that the particle moves along in the form 𝑎𝑥+𝑏𝑦=𝑐.

  • A 2 𝑥 + 3 𝑦 = 7
  • B 3 𝑥 + 2 𝑦 = 7
  • C 3 𝑥 + 2 𝑦 = 1 3
  • D 3 𝑥 − 2 𝑦 = 7
  • E 3 𝑥 − 2 𝑦 = − 1

What is the smallest value of 𝑥 during the particle’s motion? When is it reached?

  • A 𝑥 = 1 , at 𝑡=−1√2
  • B 𝑥 = 1 , at 𝑡=0
  • C 𝑥 = 0 . 5 , at 𝑡=−1√2
  • D 𝑥 = 0 . 5 , at 𝑡=0
  • E 𝑥 = − 1 , at 𝑡=1

Describe the motion from 𝑡=−1√2 to 𝑡=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 (−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,−1).
  • CThe particle starts at (2,0.5), goes right and downward to (1,2), and then goes back left and upward to (3,−1).
  • DThe particle starts at (1,2), goes left and upward to (2,1), and then goes back right and downward to (3,−1).
  • EThe particle starts at (0.5,2), goes right and downward to (2,1), and then goes back left and upward to (−1,3).

Give the parameters 𝑥=𝑓(𝑡), 𝑦=𝑔(𝑡) that describe the same motion but on an interval starting at 𝑡=0 instead of −1√2. In what interval does 𝑡 lie?

  • A 𝑥 = 𝑡 − 1 √ 2 + 1 , 𝑦 = 𝑡 − 1 √ 2 + 1 , the interval 0≤𝑡≤√2+1√2
  • B 𝑥 = 2 𝑡 + 1 √ 2 − 1 , 𝑦 = − 3 𝑡 + 1 √ 2 − 1 , the interval 0≤𝑡≤√2+1√2
  • C 𝑥 = 2 𝑡 + 1 √ 2 + 1 , 𝑦 = − 3 𝑡 + 1 √ 2 + 1 , the interval 0≤𝑡≤√2+1√2
  • D 𝑥 = 𝑡 + 1 √ 2 + 1 , 𝑦 = 𝑡 + 1 √ 2 + 1 , the interval 0≤𝑡≤√2+1√2
  • E 𝑥 = 2 𝑡 − 1 √ 2 + 1 , 𝑦 = − 3 𝑡 − 1 √ 2 + 2 , the interval 0≤𝑡≤√2+1√2

Q5:

Convert the parametric equations 𝑥=3𝑡cos and 𝑦=3𝑡sin to rectangular form.

  • A 𝑥 − 𝑦 = 9
  • B 𝑦 = 3 𝑥 s i n
  • C 𝑥 + 𝑦 = 3
  • D 𝑥 + 𝑦 = 9
  • E 𝑥 = 3 𝑦 c o s

Q6:

Convert the rectangular equation 𝑥+𝑦=25 to parametric form.

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

Q7:

Convert the parametric equations 𝑥=𝑡+2 and 𝑦=3𝑡−1 to rectangular form.

  • A 𝑥 = 𝑦 + 2
  • B 𝑥 = 𝑦 + 1 3 + 2
  • C 𝑥 = 𝑦 + 1 3 + 2
  • D 𝑥 = 3 𝑦 − 1
  • E 𝑥 = 𝑦 + 1 3

Q8:

Convert the rectangular equation 𝑥−𝑦=9 to parametric form.

  • A 𝑥 = 3 𝑡 , 𝑦 = 3 𝑡 c o s h s i n h
  • B 𝑥 = 9 𝑡 , 𝑦 = 9 𝑡 c o s h s i n h
  • C 𝑥 = 3 𝑡 , 𝑦 = 3 𝑡 c o s s i n
  • D 𝑥 = 9 𝑡 , 𝑦 = 9 𝑡 s i n c o s
  • E 𝑥 = 3 𝑡 , 𝑦 = 3 𝑡 s i n h c o s h

Q9:

Consider the points 𝐴=(−1,1) and 𝐵=(4,2). Parameterize the segment 𝐴𝐵, where 0≤𝑡≤1.

  • A 𝑥 = 1 − 𝑡 , 𝑦 = 5 𝑡 − 1
  • B 𝑥 = 𝑡 + 1 , 𝑦 = 5 𝑡 − 1
  • C 𝑥 = 5 𝑡 + 1 , 𝑦 = 1 − 𝑡
  • D 𝑥 = 5 𝑡 + 1 , 𝑦 = 1 + 𝑡
  • E 𝑥 = 5 𝑡 − 1 , 𝑦 = 𝑡 + 1

Q10:

Convert the parametric equations 𝑥=2𝑡+1 and 𝑦=𝑡−4 to the rectangular form.

  • A 𝑦 = 𝑥 − 9 2
  • B 𝑦 = 2 𝑥 + 1
  • C 𝑦 = 𝑥 + 9 2
  • D 𝑦 = 𝑥 − 4
  • E 𝑦 = 𝑥 − 1 2

Q11:

Convert the rectangular equation (𝑥+3)+(𝑦+5)=9 to parametric form.

  • A 𝑥 = 3 ( 𝑡 ) + 3 𝑦 = 3 ( 𝑡 ) + 5 c o s a n d s i n
  • B 𝑥 = 3 ( 𝑡 ) − 3 𝑦 = 3 ( 𝑡 ) − 5 c o s a n d s i n
  • C 𝑥 = 3 ( 𝑡 ) 𝑦 = 3 ( 𝑡 ) c o s a n d s i n
  • D 𝑥 = 9 ( 𝑡 ) + 3 𝑦 = 9 ( 𝑡 ) + 5 c o s a n d s i n
  • E 𝑥 = 9 ( 𝑡 ) − 3 𝑦 = 9 ( 𝑡 ) − 5 c o s a n d s i n

Q12:

Convert the parametric equations 𝑥=12𝑡ln and 𝑦=3𝑡 to rectangular form.

  • A 𝑦 = 1 2 𝑒
  • B 𝑦 = 1 2 𝑒
  • C 𝑦 = 3 8 𝑒
  • D 𝑦 = 6 𝑒
  • E 𝑦 = 3 𝑒

Q13:

Convert the parametric equations 𝑥=√𝑡 and 𝑦=5𝑡+4𝑡 to rectangular form.

  • A 𝑦 = 5 𝑥 + 4 √ 𝑥
  • B 𝑦 = 5 𝑥 + 5 𝑥
  • C 𝑦 = 5 𝑥 + 4 𝑥
  • D 𝑦 = 5 𝑥 − 4 𝑥
  • E 𝑦 = 5 𝑥 + 4 𝑥

Q14:

Convert the parametric equations 𝑥=(𝑡)cos and 𝑦=(𝑡)sin to rectangular form.

  • A 𝑦 = ( 𝑥 ) s i n
  • B 𝑥 + 𝑦 = 1
  • C 𝑥 = ( 𝑦 ) c o s
  • D 𝑥 + 𝑦 = 1
  • E 𝑥 − 𝑦 = 1

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