Conic Sections  (Video Clip--the animations will not be as smooth as in DPGraph live)

If it is not already on your hard drive, you will need to download the free DPGraph Viewer to view the DPGraph Presentations linked to on this page.

Here we investigate the graphs in the plane of equations of the form

Ax² + By² + Cx + Dy + E = 0.

We will be looking at circles, ellipses, hyperbolas, parabolas, and sometimes a line or intersecting lines.

The pictures on this page illustrate intersections in space of the graphs of planes whose equations take the form

ax + by + cz = d

and the cone whose equation is

z² = x² + y².  Animated DPGraph Intersections

In this DPGraph Presentation the initial values for a, b, c, and d in the graph of the plane are

a = 0, b = 0, c = 1, and d = 2.  The intersection of this plane with the cone would be the circle whose equation is

Click on the word "scrollbar" at the top of the page in the DPGraph Presentation, click in the small white circle beside "D variable", and then click the "OK" button.  Now by moving the slider up and down the scrollbar you can change the value of d.  Clicking in the "up" or "down" symbols at the top and bottom of the scrollbar will change the value of d in increments of 0.25 and allow you to return d to its initial value of 2.  You can also "grab" the slider and change the value of d by increments other than 0.25.  Changing d will cause the radius of the circle of intersection of the plane and cone to change.

Use the scrollbar to return the value of d to about 2 and then click "scrollbar" at the top of the page again.  This time click beside "A variable".  This will activate a and deactivate d (leaving the d value as you left it).  The initial value of a will be 0.  Use the scrollbar to change the value of a.  This will "tilt" the plane and cause the intersection to be an ellipse.  You can use your arrow keys to change the view by rotating the graph.  The pictures below show the ellipse that results when  d = 2 and a = -0.5.  The equation of this ellipse in the xy-coordinate plane would be

Changing the value of a to -1 in the DPGraph Presentation with d = 2 produces a parabola whose equation in the xy-coordinate plane would be

The pictures below show this parabola that results when d = 2 and a = -1.

If we move down the scrollbar in the DPGraph Presentation until the value of a is -2 the intersection of the plane and the cone will be a hyperbola whose equation in the xy-coordinate plane will be

The pictures below show this hyperbola (and its asymptotes) that results when d = 2 and a = -2.

Additional hyperbolas could be produced with a values of -4 and 4 for example.

Bring the value of a back to -1 (parabola) in the DPGraph Presentation.  Click on "scrollbar" at the top of the page and activate d again.  Change the d-value to 0.  The resulting intersection is pictured below.

In the DPGraph Presentation click on "scrollbar" at the top of the page and activate c.  Change the c-value to 0.  In the pictures below the values of b, c, and d are 0 and the value of a is -1.

Try making other changes in a, b, c, and d in the DPGraph Presentation and see what you get.  You can earn up to 5 bonus points (one for each case) by indicating what you get for various combinations of values for a, b, c, and d not investigated on this page.

Here is a Maple Worksheet relating to the graphs displayed on this page.

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        Lane Vosbury, Mathematics, Seminole State College   email:  vosburyl@seminolestate.edu

        This page was last updated on 08/21/14          Copyright 2002          webstats