Education Technology

# Conics as a Locus of Points

Math: PreAlgebra: Geometry
6-8
90 Minutes
TI Connect™
TI-83 Plus Family
TI-84 Plus
TI-84 Plus Silver Edition
The following equipment is required for this activity:
• TI-Presenter™

• LCD Projector
• Two screens
Lessons

# Conics as a Locus of Points

Activity Overview

Students investigate the definition of a parabola through one of its geometric definitions. They study conic sections. They examine an ellipse as a locus of points such that the sum of distances from the foci to the traced path is constant.

Before the Activity

Install the Cabri Jr.™ App on the students' graphing calculators using one of these two methods:

• TI-Connect™,  a TI Connectivity Cable, and the Unit-to-Unit Link Cable
• TI-Navigator™  "send to class" feature
• See the attached PDF file for detailed instructions for this activity
• Print pages 1 - 6 from the attached PDF file for your class
• During the Activity

Distribute the pages to your class.

Exploration 1:

• Draw a segment to connect a line (directrix) with a point (focus) not on it
• Construct a perpendicular to the directrix at the point D where the segment intersects it
• Construct a perpendicular bisector of the original segment
• Mark Point P as the point of intersection of the two perpendiculars
• Draw a circle using P as center P passing through the focus
• Drag point D along the directrix
• Observe that the path traced by point P is a parabola

• Exploration 2:
• Draw a circle
• Draw a segment connecting a point A on the circle to a point B within the circle
• Construct a perpendicular bisector of the segment
• draw a line to connect the point A to the center of the circle
• Label the point of intersection of the radius and the bisector as P
• Drag the point A on the circle along the circumference
• Observe that the path traced by P is an Ellipse
• Drag point B outside the circle and observe the path traced is a hyperbola

• Exploration 3:
• Draw a circle and mark a point B outside the circle
• Draw a second circle with a point on the first circle as its center, and passing through B
• Drag the center of the second circle around the circumference of the first circle
• Observe that the path traced by point B is a limacon with a loop when the point is outside the first circle, a limacon without a loop if it is inside the circle, and a cardioid if it is on the circle
• After the Activity

Review student results:

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary
Math: PreAlgebra: Geometry
6-8
90 Minutes
TI Connect™
TI-83 Plus Family
TI-84 Plus
TI-84 Plus Silver Edition
The following equipment is required for this activity:
• TI-Presenter™

• LCD Projector
• Two screens
Lessons
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