# 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 to find a parametrization of the part of the hyperbola that contains the point .

• A
• B
• C
• D
• E

Q2:

The first figure shows the graphs of and that parameterize the unit circle for . What do the two functions graphed in the second figure parameterize? • AThe unit circle
• BThe square on , , ,
• CThe square on , , ,
• DThe square on , , ,

Q3:

Consider the points and .

What is the length of ?

Find a parameterization of the segment over .

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• B
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• E

Find and so that , parameterizes over .

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• E

Using the functions above for , what is the distance between the point and the point ?

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• B
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• E

The parameterization of above is an example of an {arc-length parameterization} of a plane curve. Find an arc-length parameterization , of , where and the parameter starts at . Give the interval used.

• A on
• B on
• C on
• D on
• E on

Q4:

A particle moves along the curve given by the parametric equations , with .

At which point is the particle located when ? Give your answer to one decimal place, if necessary.

• A
• B
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• D
• E

At which points is the particle located when and ? Give your answers to one decimal place, if necessary.

• A and
• B and
• C and
• D and
• E and

Find an equation for the line that the particle moves along in the form .

• A
• B
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• D
• E

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

• A, at
• B, at
• C, at
• D, at
• E, at

Describe the motion from to in terms of position on the line.

• AThe particle starts at , goes left and upward to , and then goes back right and downward to .
• BThe particle starts at , goes left and upward to , and then goes back right and downward to .
• CThe particle starts at , goes right and downward to , and then goes back left and upward to .
• DThe particle starts at , goes left and upward to , and then goes back right and downward to .
• EThe particle starts at , goes right and downward to , and then goes back left and upward to .

Give the parameters , that describe the same motion but on an interval starting at instead of . In what interval does lie?

• A, , the interval
• B, , the interval
• C, , the interval
• D, , the interval
• E, , the interval

Q5:

Convert the parametric equations and to rectangular form.

• A
• B
• C
• D
• E

Q6:

Convert the rectangular equation to parametric form.

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• C
• D
• E

Q7:

Convert the parametric equations and to rectangular form.

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• B
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• D
• E

Q8:

Convert the rectangular equation to parametric form.

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• D
• E

Q9:

Consider the points and . Parameterize the segment , where .

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• E

Q10:

Convert the parametric equations and to the rectangular form.

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• B
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• D
• E

Q11:

Convert the rectangular equation to parametric form.

• A
• B
• C
• D
• E

Q12:

Convert the parametric equations and to rectangular form.

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• B
• C
• D
• E

Q13:

Convert the parametric equations and to rectangular form.

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• B
• C
• D
• E

Q14:

Convert the parametric equations and to rectangular form.

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• B
• C
• D
• E

Q15:

Convert the rectangular equation to parametric form.

• A
• B
• C
• D
• E