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).

  • A𝑥=5𝑡,𝑦=9𝑡coshsinh
  • B𝑥=−5𝑡,𝑦=−9𝑡coshsinh
  • C𝑥=25𝑡,𝑦=81𝑡coshsinh
  • D𝑥=−5𝑡,𝑦=9𝑡coshsinh
  • E𝑥=5𝑡,𝑦=−9𝑡coshsinh

Q2:

The first figure shows the graphs of cos2𝜋𝑡 and sin2𝜋𝑡 that 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.

  • A𝑥=4(𝑡+1),𝑦=3(𝑡+1)
  • B𝑥=𝑡+4,𝑦=𝑡+3
  • C𝑥=4𝑡+1,𝑦=3𝑡+1
  • D𝑥=3𝑡+1,𝑦=4𝑡+1
  • E𝑥=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.

  • A𝑓(𝑡)=4𝑡+1,𝑔(𝑡)=3𝑡+1
  • B𝑓(𝑡)=𝑡+4,𝑔(𝑡)=𝑡+3
  • C𝑓(𝑡)=3𝑡+1,𝑔(𝑡)=4𝑡+1
  • D𝑓(𝑡)=𝑡+45,𝑔(𝑡)=𝑡+35
  • E𝑓(𝑡)=45𝑡+1,𝑔(𝑡)=35𝑡+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))?

  • A𝑠−1
  • B𝑠+1
  • C𝑠
  • D𝑠+2
  • E𝑠−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.

  • A𝑥=1213𝑡+1,𝑦=513𝑡+1 on 5≤𝑡≤13
  • B𝑥=1213𝑡+1,𝑦=513𝑡+1 on 0≤𝑡≤13
  • C𝑥=513𝑡+1,𝑦=1213𝑡+1 on 0≤𝑡≤13
  • D𝑥=𝑡+1213,𝑦=513𝑡 on 0≤𝑡≤13
  • E𝑥=513𝑡+1,𝑦=1213𝑡+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.

  • A2𝑥+3𝑦=7
  • B3𝑥+2𝑦=7
  • C3𝑥+2𝑦=13
  • D3𝑥−2𝑦=7
  • E3𝑥−2𝑦=−1

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

  • A𝑥=1, at 𝑡=−1√2
  • B𝑥=1, at 𝑡=0
  • C𝑥=0.5, at 𝑡=−1√2
  • D𝑥=0.5, at 𝑡=0
  • E𝑥=−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?

  • A𝑥=𝑡−1√2+1, 𝑦=𝑡−1√2+1, the interval 0≤𝑡≤√2+1√2
  • B𝑥=2𝑡+1√2−1, 𝑦=−3𝑡+1√2−1, the interval 0≤𝑡≤√2+1√2
  • C𝑥=2𝑡+1√2+1, 𝑦=−3𝑡+1√2+1, the interval 0≤𝑡≤√2+1√2
  • D𝑥=𝑡+1√2+1, 𝑦=𝑡+1√2+1, the interval 0≤𝑡≤√2+1√2
  • E𝑥=2𝑡−1√2+1, 𝑦=−3𝑡−1√2+2, the interval 0≤𝑡≤√2+1√2

Q5:

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

  • A𝑥−𝑦=9
  • B𝑦=3𝑥sin
  • C𝑥+𝑦=3
  • D𝑥+𝑦=9
  • E𝑥=3𝑦cos

Q6:

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

  • A𝑥=𝑡,𝑦=25𝑡sincos
  • B𝑥=5𝑡,𝑦=5𝑡cossin
  • C𝑥=5𝑡,𝑦=5𝑡cossin
  • D𝑥=𝑡,𝑦=𝑡sincos
  • E𝑥=25𝑡,𝑦=25𝑡sincos

Q7:

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

  • A𝑥=𝑦+2
  • B𝑥=𝑦+13+2
  • C𝑥=𝑦+13+2
  • D𝑥=3𝑦−1
  • E𝑥=𝑦+13

Q8:

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

  • A𝑥=3𝑡,𝑦=3𝑡coshsinh
  • B𝑥=9𝑡,𝑦=9𝑡coshsinh
  • C𝑥=3𝑡,𝑦=3𝑡cossin
  • D𝑥=9𝑡,𝑦=9𝑡sincos
  • E𝑥=3𝑡,𝑦=3𝑡sinhcosh

Q9:

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

  • A𝑥=1−𝑡,𝑦=5𝑡−1
  • B𝑥=𝑡+1,𝑦=5𝑡−1
  • C𝑥=5𝑡+1,𝑦=1−𝑡
  • D𝑥=5𝑡+1,𝑦=1+𝑡
  • E𝑥=5𝑡−1,𝑦=𝑡+1

Q10:

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

  • A𝑦=𝑥−92
  • B𝑦=2𝑥+1
  • C𝑦=𝑥+92
  • D𝑦=𝑥−4
  • E𝑦=𝑥−12

Q11:

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

  • A𝑥=3(𝑡)+3𝑦=3(𝑡)+5cosandsin
  • B𝑥=3(𝑡)−3𝑦=3(𝑡)−5cosandsin
  • C𝑥=3(𝑡)𝑦=3(𝑡)cosandsin
  • D𝑥=9(𝑡)+3𝑦=9(𝑡)+5cosandsin
  • E𝑥=9(𝑡)−3𝑦=9(𝑡)−5cosandsin

Q12:

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

  • A𝑦=12𝑒
  • B𝑦=12𝑒
  • C𝑦=38𝑒
  • D𝑦=6𝑒
  • E𝑦=3𝑒

Q13:

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

  • A𝑦=5𝑥+4√𝑥
  • B𝑦=5𝑥+5𝑥
  • C𝑦=5𝑥+4𝑥
  • D𝑦=5𝑥−4𝑥
  • E𝑦=5𝑥+4𝑥

Q14:

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

  • A𝑦=(𝑥)sin
  • B𝑥+𝑦=1
  • C𝑥=(𝑦)cos
  • D𝑥+𝑦=1
  • E𝑥−𝑦=1

Q15:

Convert the rectangular equation (𝑥+2)−(𝑦+5)=16 to parametric form.

  • A𝑥=4(𝑡)−2𝑦=4(𝑡)−5coshandsinh
  • B𝑥=4(𝑡)𝑦=4(𝑡)coshandsinh
  • C𝑥=16(𝑡)−2𝑦=16(𝑡)−5coshandsinh
  • D𝑥=4(𝑡)−2𝑦=4(𝑡)−5cosandsin
  • E𝑥=4(𝑡)+2𝑦=4(𝑡)+5cosandsin

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