Education Technology

Conics as a Locus of Points

Published on 06/09/2008

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.

    Follow the Activity procedures:

    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