Erik D. Jacobson

Final 2 | Home

In this exploration we continue to explore the equations of conics in polar form. In particular, we examine the axis of symmetry for a conic and continue to explore how the equation changes the shape of the conic that is graphed. (See this page for more discussion.)

The equation we explored below is:

*

The exploration is captured in the following movie which has three variants of (*) graphed. These variants are:

graphed in green and

which is graphed in blue.

The red line in the graph is given by  .

When k = -3, both the green and blue ellipse are horizontally oriented, but because k is negative, the right focus of the green ellipse is at the origin whereas the left focus of the blue ellipse is at the origin.

As k increases to -2, the orientation of both ellpses pass through 180 degrees counterclockwise. Now the left focus of the green ellipse and the right focus of the blue ellipse are at the origin.  In this period you can also see that the green ellipse has increased in size whereas the blue ellipse has shrunk. The reason for this change of scale is that the parameter k or its inverse is multiplied by (*) the size of the ellipse generated changes corresponding to the value of k.

When k = -1, the ellipses are now the same size, but still have opposite orientation.  As k increases to 0, we see that the green ellipse becomes a parabola and then a hyperbola, the transition happening around k = 0.5.  As k approaches 0 from below, the green hyperbola approaches parallel lines.

As k increases to 1, about k = .5 we can again see the gree hyperbola become a parabola and then an ellipse. Notice that now both the blue and green ellipses have the same orientation in the sense that both have their right foci at the origin.  When k = 1, the ellipses coincide.

In the last 2 units of k's increase from 1 to 3, the green ellipse continues to shrink and the blue ellipse continues to increase in size.  By symmetry, we might expect the blue ellipse to eventually turn into a hyperbola, but this is not the case (see this page for more details).