# Activities

• • • ##### Subject Area

• Math: Precalculus: Parametric Equations

• ##### Author College

60 Minutes

• ##### Device
• TI-89 / TI-89 Titanium
• TI-92 Plus / Voyage™ 200

Elimination

#### Activity Overview

This activity uses geometric examples to illustrate the method of elimination to obtain a Cartesian equation using the Computer Algebra System CAS tool.

#### Before the Activity

• See the attached PDF file for detailed instructions for this activity
• Print pages 1 - 6 from the attached PDF file for the class
• #### During the Activity

Distribute the pages to the class.

• In a vectorial plane, start at a certain point, and reach a certain point taking some steps of a particular vector
• Observe the line which is generated over R
• Recognize the fact that the parametric equations of the line are the algebraic translation of the geometric description of the line
• Find points on the line for two values of the 'vector step'
• Alternatively, decide whether a given point lies on the line
• Understand that a point lies on the line if the system has a solution for the 'vector-step'
• Verify the fact by solving the system
• Realize that finding the Cartesian equations of a line is equivalent to finding the conditions for the coordinates of a general point in order that the system would have a solution for the 'vector-step'
• Eliminate the step-parameter and solve the system
• Repeat the steps by forming a plane over R when steps of two different vectors are taken to reach a point

• Enter the equations of the circle and the line
• Find the points of intersection and determine the equations of the two solutions
• Draw the curve for the parametric equations
• Eliminate the parameter 't' from the system of the associated curves, and determine the Cartesian equation of the Strophoid
• Translate the double point of the Strophoid to the origin of the coordinate system
• Intersect the curve with the line y = tx
• Obtain the parametric representation for the whole strophoid
• #### After the Activity

Students understand that with the help of CAS, parameters can be eliminated directly without the use of determinants.

Review student results.

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary