Defining a Parabola with Three PointsDate: 05/09/2009 at 01:45:31 From: Junichi Subject: Do 3 points determining a parabola if 1 point is the vertex If we have 3 distinct points, and we know that one point is the vertex (of a parabola), is there only one parabola (any direction) that passes through the 3 points? I think there is one and only one parabola that passes through the 3 points, but I'm not sure. This is not homework by the way. I just came up with this crazy thought while learning parabolas in school. Note: We know one point is the vertex, it's not like the "5 point locate one parabola" theorem. Since the parabola can be going towards any direction, hence it doesn't have to apply to y=ax^2+bx+c. Therefore the only way to prove this is by non-coordinate geometry? Note: If we know that the parabola satisfies y=ax^2+bx+c (or y=a(x- h)^2+k) then we only need 2 points. A vertex (h,k) and a point (x,y). I tried to find the location of the axis of symmetry first. If we can locate it, and if it's unique, then we can reflect the points besides the vertex, then we'll have 5 points. And we know that 5 point determine a parabola (2 parabolas intersect at most 4 points). Date: 05/10/2009 at 05:17:49 From: Doctor Jacques Subject: Re: Do 3 points determining a parabola if 1 point is the vertex Hi Junichi, First, note that, although you need 5 points to determine a general conic, 4 points will suffice if you know that the conic is a parabola. This is because a parabola has only one asymptotic direction, which means that the discriminant of the second degree terms is 0. This gives you an additional equation, which removes one "degree of freedom" (this is strictly correct, you will see below what I mean). If, in addition to that, you know that one (identified) point is a vertex, you have another equation, because the tangent at the vertex is perpendicular to the asymptotic direction, and you need one point less. We may choose the vertex (A) as the origin of the axes. The general equation of a parabola (not vertical) through the origin is: (y - mx)^2 + ay + bx = 0 [1] where m is the slope of the asymptotic direction. The slope of the tangent at the origin is (-b/a). If the origin is the vertex, this slope must be perpendicular with m, and we have: (-b/a) = (-1/m) a = mb [2] and, by substituting into [1], you get the equation: (y - mx)^2 + b(my + x) = 0 [3] Each of the other two points (B and C) will give you an equation, and you can solve those two equations for the unknowns m and b. The point is that you will get a cubic equation in m (you can see this, for example, by choosing one of the coordinate axes as the line through AB). This equation may have up to three real roots for m, and each such root will give you one value for b. This means that you can actually have up to 3 parabolas through the given points. For example, assume that the points are: A (0,0) B (0,1) C (1,3) If you work through the substitutions, you will find that m is given by the equation: m^3 - 6m^2 + 6m - 1 = 0 [4] and this equation has three real roots: {0.20871215252208, 1.0, 4.79128784747792}. Each of these roots gives you the equation of a parabola through A, B, C : (y - 0.20871215252208x)^2 - y - 4.79128784747792x = 0 (x - y)^2 - y - x = 0 (y - 4.79128784747792x)^2 - y - 0.20871215252208x = 0 (The symmetry between the first and third equations is accidental, it is related to the fact that the equation [4] is self-reciprocal; this is not typical of the general case). You can see the three parabolas on: Please feel free to write back if you require further assistance. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ Date: 05/20/2009 at 01:47:40 From: Junichi Subject: Do 3 points determining a parabola if 1 point is the vertex Hi, First of all thank you for helping me. But can you clarify what's the asymptotic direction you're referring to? Is it the direction of the axis of symmetry? Or the directrix? Thanks again, Junichi Date: 05/20/2009 at 04:55:54 From: Doctor Jacques Subject: Re: Do 3 points determining a parabola if 1 point is the vertex Hi again Junichi, An asymptotic direction on a curve is the limit of the ratio y/x (the slope of a line through the origin) as the point on the curve goes to infinity. If the curve has an asymptote, then the asymptotic direction is the slope of the asymptote. For example, a hyperbola has two asymtoptes, and the asymptotic directions are the slopes of those asymptotes. In a parabola, there is only one asymptotic direction, and there is no corresponding asymptote. The asymptotic direction is the direction of the axis of symmetry. For a general conic, you can compute the asymptotic directions by keeping only the second degree terms, and solving the corresponding homogeneous equation for y/x. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ |
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