Lesson: Conversion between Parametric and Rectangular Equations Mathematics • Higher Education

In this lesson, we will learn how to convert from the parametric form of an equation to its equivalent rectangular form and vice versa.

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12:31

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Lesson Worksheet

Q1:

Use the fact that coshsinhπ‘₯βˆ’π‘₯=1 to find a parametrization of the part of the hyperbola π‘₯25βˆ’π‘¦81=1 that contains the point (βˆ’5,0).

Q2:

The first figure shows the graphs of cos2πœ‹π‘‘ and sin2πœ‹π‘‘ that parameterize the unit circle for 0≀𝑑≀1. What do the two functions graphed in the second figure parameterize?

Q3:

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

What is the length of 𝐴𝐡?

Find a parameterization of the segment 𝐴𝐡 over 0≀𝑑≀1.

Find 𝑓 and 𝑔 so that π‘₯=𝑓(𝑑), 𝑦=𝑔(𝑑) parameterizes 𝐴𝐡 over 0≀𝑑≀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 𝐴𝐡 above is an example of an {arc-length parameterization} of a plane curve. Find an arc-length parameterization π‘₯=𝑓(𝑑), 𝑦=𝑔(𝑑) of 𝐴𝐢, where 𝐢=(13,6) and the parameter starts at 𝑑=0. Give the interval used.

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