Straight Lines and Conic SectionsDate: 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 http://en.wikipedia.org/wiki/Eccentricity_(mathematics) 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 or (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 or (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 http://mathforum.org/dr.math/ |
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