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).