Re: tangents and their respective equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg31933] Re: [mg31846] tangents and their respective equations*From*: "Philippe Dumas" <dumasphi at noos.fr>*Date*: Thu, 13 Dec 2001 01:08:36 -0500 (EST)*References*: <200112071056.FAA10550@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi Manuel, Clearly your problem is an ill-posed problem : it is obviously impossible to derive the equation of one curve given its tangent at one point ! One reason (among others) is that there are infinitely many curves being tangent to one line at one particular point. This said, there is an interesting question related to you (slightly modified) problem. Namely, given a family of straight lines defined by an equation f(m,x)=0 depending on a free parameter m (e.g. f(m,x) = 2*m*x - m^2*y + m -1 = 0), what is the curve being envelopped by this family of lines (i.e. what is the curve everywhere tangent to one particular line of the family ) ? The answer is rather straigthfoeward (at least in principle) : the equation of the curve is obtained by eliminating the free parameter m between f(m,x) = 0 and D[f(m,x),m] = 0 where D stands for the partial derivative of f vs. m. In the given example, the solution is to eliminate m between: 2*m*x - m^2*y + m -1 = 0 (1) and 2*x -2*m*y +1 = 0 (2) introducing this value in (1) Obviously one has to take care of problems arising because of divisions by zero (here at y=0) I hope this will be helpful. Philippe Dumas 99, route du polygone 03 88 84 67 80 67100 Strasbourg ----- Original Message ----- From: "Manuel Avalos" <manuel at voicenet.com> To: mathgroup at smc.vnet.net Subject: [mg31933] [mg31846] tangents and their respective equations > > Hi everyone: > > I am trying to find the equation of a curve given its tangent and an > arbitrary point. Is there a mathematica solution for this? > For example: If I give you the tangent to a curve: -4+11x at x =2, > Is it possible to derive the equation of the curve from that tangent? > Thanks for whatever. > Manuel > >

**References**:**tangents and their respective equations***From:*"Manuel Avalos" <manuel@voicenet.com>