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
Follow the Activity procedures:
After the Activity
Students understand that with the help of CAS, parameters can be eliminated directly without the use of determinants.As a class, discuss questions that appeared to be more challenging
Re-teach concepts as necessary
Review student results.