Start Practicing
Worksheet: Parametric Equations of Plane Curves
Q1:
A particle following the parameterization , of the unit circle starts at and moves counterclockwise. At what values of is the particle at ? Give exact values.
- A
- B
- C
- D
- E
Q2:
The diagram shows a parabola that is symmetrical about the -axis and whose vertex is at the origin. It can be described by the parametric equations and , , where is a positive constant. The focus of the parabola is the point , and the directrix is the line with the equation .
Find a pair of parametric equations that describe the parabola whose focus is the point and whose directrix is the line . Include the parameter range.
- A , ,
- B , ,
- C , ,
- D , ,
- E , ,
Q3:
The first figure shows the graphs of and , which 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 , , ,
Q4:
Use the fact that to find a parametrization of the part of the hyperbola that contains the point .
- A
- B
- C
- D
- E
Q5:
A particle following the parameterization , of the unit circle starts at and moves counterclockwise. At what values of is the particle at ? Give exact values.
- A , where is a nonnegative integer
- B , where is a nonnegative integer
- C , where is a nonnegative odd number
- D , where is a nonnegative integer
- E , where is a nonnegative integer
Q6:
Consider the points and .
What is the length of ?
Find a parameterization of the segment over .
- A
- B
- C
- D
- E
Find and so that , parameterizes over .
- A
- B
- C
- D
- E
Using the functions above for , what is the distance between the point and the point ?
- A
- B
- C
- D
- 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