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Straight Lines and Conic Sections

Date: 10/14/2008 at 14:18:29
From: Anonymous
Subject: pair of straight lines as a conic section

Dear Dr. Math,

Is a pair of intersecting straight lines a conic section?  If so, what
is its eccentricity value?  I heard some say that its eccentricity
value tends to infinity, but my math professor disagrees.  Please help me.

Date: 10/16/2008 at 14:54:40
From: Doctor Vogler
Subject: Re: pair of straight lines as a conic section

Hi Nony,

Thanks for writing to Dr. Math.  The eccentricity of a conic section 
is usually defined for each conic section, and particularly only for 
the nondegenerate conics.  So the simplistic answer is that the
eccentricity of a pair of intersecting lines is not defined.

Of course, that leaves unanswered the question "Is there a reasonable
way to define it?"  For example, "infinity" is often a reasonable
answer for "undefined" things; sometimes a nonzero quantity divided by
zero is taken to be infinity, which makes a certain sense in terms of
limits, and in certain other special cases, although this is not
always meaningful.  Certainly, if someone gives you a (positive)
finite eccentricity, you could say what type of nondegenerate conic
has that eccentricity, so the natural guess at the eccentricity of a
degenerate conic would be infinity, the one positive "number" that was
left out of the list.  That also jives with statements you'll find on
places like

  Wikipedia: Eccentricity 

that eccentricity is a measurement of distance from circular, and
eccentricity increases as curvature decreases, since lines have zero
curvature.  But actually that's an over-simplification, since all
circles have zero eccentricity, but larger circles have smaller curvature.

For another thing, the similarity of the definitions of eccentricity
for ellipses and hyperbolas lead one to think that maybe there's a
general formula for a general conic section in quadratic form

  ax^2 + bxy + cy^2 + dx + ey + f = 0

which would give the correct values for circles, ellipses, parabolas,
and hyperbolas, and then we need only evaluate this formula for the
degenerate cases to find out what their eccentricity is. 

Unfortunately, things aren't quite so nice.  But I'll demonstrate how
close it gets, and how things break down for degenerate conics.

After a rotation and a translation, you can put most conics (but not
parabolas, for example) into the simplified form

  ax^2 + by^2 = c.

Applying to our formula for eccentricity of an ellipse or a hyperbola,
we find that

  1) if 0 < b < a and 0 < c, then e = sqrt((b-a)/b)
  2) if b < 0 < a and 0 < c, then e = sqrt((b-a)/b)
  3) if a < b < 0 and c < 0, then e = sqrt((b-a)/b)
  4) if a < 0 < b and c < 0, then e = sqrt((b-a)/b)

which looks promising, but

  5) if 0 < a < b and 0 < c, then e = sqrt((a-b)/a)
  6) if a < 0 < b and 0 < c, then e = sqrt((a-b)/a)
  7) if b < a < 0 and c < 0, then e = sqrt((a-b)/a)
  8) if b < 0 < a and c < 0, then e = sqrt((a-b)/a).

Well, okay, so there's not one formula; there's two.  That's a little
annoying, and one might wonder which to use in unknown cases (and, of
course, I have left out parabolas), but let's work with what we have.

Now suppose that b = 1 is positive, and a = -m^2 is negative, so that
our equation is

  y^2 - (mx)^2 = c


  (y - mx)(y + mx) = c.

If c > 0, then we have a hyperbola with eccentricity sqrt(1 + m^(-2)),
independent of c.  If c < 0, then we have a hyperbola with 
eccentricity sqrt(1 + m^2), independent of c.  If c = 0, then we have
a pair of intersecting lines.  So what should we consider its
eccentricity to be?

Well, it's right there on the boundary where the eccentricity changes
from one constant to another.  The only reasonable choice seems to be
undefined, but *not* because it's infinity.  At least, it usually
changes from one constant to another.  But what if m = 1?

If we consider the conic section

  y^2 - x^2 = c


  (y - x)(y + x) = c

then we find that we have a hyperbola with eccentricity sqrt(2) for
every value of c *except* zero.  When c = 0, we have a pair of
intersecting lines.  Naturally, we should expect this pair of
intersecting lines to have eccentricity sqrt(2), right?  Not infinity.

Anyway, in summary I would conclude that eccentricity is only defined
for nondegenerate conic sections, and since it depends on which
parameters are larger (where is the "major" axis and the "minor"
axis), it doesn't generalize nicely to the degenerate conic sections,
though if it did, it would seem that the degenerate conic sections
would have eccentricity values in common with some of the 
nondegenerate ones.  For example,

  x^2 + y^2 = r^2

has eccentricity 0, so the single point (when r = 0) should have
eccentricity 0.  And

  y = mx^2 + c

has eccentricity 1, so the single line (when m = 0) should have
eccentricity 1.  And I already argued about the case of two
intersecting lines, which are (usually) right on the border between
two different eccentricity values.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum 
Associated Topics:
College Conic Sections/Circles
High School Conic Sections/Circles

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