Janet Kaplan

Conic Sections

Graphs of the form x^{2} + nxy + y^{2} are called Conic
Sections. They include circles, ellipses and hyperbolas.

For this investigation we
will vary the coefficient of the xy term, n, and explore what results we get.

We start out with the basic
equation of a circle, say
x^{2 }+ y^{2} = 4. We then add a central term, xy, and examine the results as
we vary its coefficient.

What will happen if we add
the central term, xy? Let's set n = 1 and see what happens.

x^{2} + xy + y^{2}

How about if we set n equal
to -1? So, x^{2}
- xy + y^{2} = 4:

Adding that central term
changes the circle into an ellipse (assuming, of course, that n does not equal
0).

But now to really see the
changes taking place with our basic circle, click here. This
will animate the function as n changes from -3 to 3.

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What do you notice? First
of all, the sign of n seems to determine the orientation of the ellipse. When
xy has a positive coefficient, an ellipse that schews to the left is formed;
when the coefficient is negative, the ellipse schews to the right.

Staying with positive
coefficients for the moment, as we increase from n = 1 to 2, the ellipse
elongates. This happens gradually at first, but by n = 1.8, the ellipse is very
long, and at n = 2, the ellipse becomes two parallel lines!

Immediately beyond the
parallel and continuing further towards n = 3, the function becomes a
hyperbola. With increasing n values, the angle of the hyperbola becomes
steeper.

Notice that all the while
the function contains the points (2,0), (-2,0), (0,2), (0,-2).

This is perhaps the only
thing that is immediately clear. What is happening to our simple circle to make
it become an ellipse, then parallel lines, and then a hyperbola?

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Let’s refer to the geometry of Conic Sections to help us
visualize what is happening.

Conic sections are defined
to be the nondegenerate curves generated by the intersections of a plane with
one or two nappes of a cone. For a plane perpendicular to the axis of the cone,
a circle is produced. For a plane which is not perpendicular to the axis and
which intersects only a single nappe, the curve produced is either an ellipse
or a parabola. The curve produced by a plane intersecting both nappes is a
hyperbola. **

** Mathworld.Wolfram.com

The graphs above are
actually a cross-section of the cone generated by the intersection of the
plane. Notice how a circle is created when the plane is perpendicular to the
axis of the cone. This occurs when n = 0, resulting in the equation

x^{2} + y^{2} = 4.

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As n increases from 0 to 2,
the circle begins to elongate into an ellipse. The plane intersecting the cone
meets it at an angle, thereby creating the ellipse. The closer to zero n is,
the closer the perpendicularity of the angle, and the smaller the eccentricity of the
ellipse. At n = 1 the angle of the plane is somewhat steep, and the eccentricity is 0.816 (0 < eccentricity < 1). The ellipse continues to elongate
as n increases towards 2, and at the exact moment that it reaches 2, the
ellipse becomes two parallel lines! It is known as a degenerate curve at this
point.

Again, using the
three-dimensional figures above as a model, we can see how this happens. Two
parallel lines occur when the plane's angle of intersection is such that it is
tangent to the left surface of one nappe and to the right surface of the other
nappe. It would be going through the center point of the inverted cones. Can
you picture it?

Continuing the plane's
ascent to an angle that is parallel to the cone's axis and perpendicular to the
base, we can see that hyperbolas are formed. The hyperbola starts out quite
flat when n is just slightly larger than 2, but becomes increasingly curved as
it moves towards and then beyond 3 and 4.

See it? At n values just
over 2, the parallel plane intersects the cone close to the center. As n
increases, the plane moves outward towards the edges of the cone, bringing the
eccentricity of the hyperbola down closer to one.

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