Lesson: Parametric Equations of Plane Curves

In this lesson, we will learn how to find the parametric equations of a plane curve.

Worksheet: Parametric Equations of Plane Curves • 6 Questions

Q1:

Use the fact that c o s h s i n h 𝑥 − 𝑥 = 1 to find a parametrization of the part of the hyperbola 𝑥 2 5 − 𝑦 8 1 = 1 that contains the point ( − 5 , 0 ) .

Q2:

A particle following the parameterization 𝑥 = ( 2 𝜋 𝑡 ) c o s , 𝑦 = ( 2 𝜋 𝑡 ) s i n of the unit circle starts at ( 1 , 0 ) and moves counterclockwise. At what values of 0 ≤ 𝑡 ≤ 4 is the particle at ( 0 , 1 ) ? Give exact values.

Q3:

The diagram shows a parabola that is symmetrical about the 𝑥 -axis and whose vertex is at the origin. It can be described by the parametric equations 𝑥 = 𝑝 𝑡 2 and 𝑦 = 2 𝑝 𝑡 , 𝑡 ∈ ℝ , where 𝑝 is a positive constant. The focus of the parabola is the point ( 𝑝 , 0 ) , and the directrix is the line with the equation 𝑥 + 𝑝 = 0 .

Find a pair of parametric equations that describe the parabola whose focus is the point 3 2 , 0 and whose directrix is the line 𝑥 = − 3 2 . Include the parameter range.

Q4:

The first figure shows the graphs of c o s 2 𝜋 𝑡 and s i n 2 𝜋 𝑡 , which parameterize the unit circle for 0 ≤ 𝑡 ≤ 1 . What do the two functions graphed in the second figure parameterize?

Q5:

A particle following the parameterization 𝑥 = 2 𝜋 𝑡 c o s , 𝑦 = 2 𝜋 𝑡 s i n of the unit circle starts at ( 1 , 0 ) and moves counterclockwise. At what values of 𝑡 ≥ 0 is the particle at ( 0 , 1 ) ? Give exact values.

Q6:

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

What is the length of 𝐴 𝐵 ?

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

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

Using the functions above for 0 ≤ 𝑠 ≤ 5 , what is the distance between the point ( 1 , 1 ) and the point ( 𝑓 ( 𝑠 ) , 𝑔 ( 𝑠 ) ) ?

The parameterization of 𝐴 𝐵 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 𝐴 𝐶 , where 𝐶 = ( 1 3 , 6 ) and the parameter starts at 𝑡 = 0 . Give the interval used.
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