Re: Reduce/Solve

*To*: mathgroup at smc.vnet.net*Subject*: [mg50060] Re: Reduce/Solve*From*: ankowar at yahoo.com (Andy Kowar)*Date*: Fri, 13 Aug 2004 05:56:20 -0400 (EDT)*References*: <200408090829.EAA03580@smc.vnet.net> <200408101002.GAA19500@smc.vnet.net> <766AE111-EABF-11D8-BBFD-000A95B4967A@mimuw.edu.pl> <opsciyqjd6iz9bcq@monster.cox-internet.com> <395198E8-EB0D-11D8-AA51-000A95B4967A@mimuw.edu.pl> <cfcr2e$475$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

I have a problem that the experts on Solve/Reduce participating in this thread might help me with. I define an ellipse in parametric form: r[t_]:={a*Cos[t],b*Sin[t]} and the tangent dr[t_]:=Evaluate[D[r[t], t]] I define a vector q={k,l}; I want to find points on the ellipse such that the tangent is parallel to a given vector: q={k,l}; I expect two two solutions. I define the equation: eq = dr[t] == lambda q; The following command sol = Simplify[Solve[eq, {t, lambda}], {a > b > 0, k > 0, l\[Element]Reals}] produces the following warning "Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information" and the following four solutions (here in the InputForm) {{lambda -> -((a*b)/Sqrt[b^2*k^2 + a^2*l^2]), t -> -ArcCos[-((a*l)/Sqrt[b^2*k^2 + a^2*l^2])]}, {{lambda -> -((a*b)/Sqrt[b^2*k^2 + a^2*l^2]), t -> -ArcCos[-((a*l)/Sqrt[b^2*k^2 + a^2*l^2])]}, {lambda -> -((a*b)/Sqrt[b^2*k^2 + a^2*l^2]), t -> ArcCos[-((a*l)/Sqrt[b^2*k^2 + a^2*l^2])]}, {lambda -> (a*b)/Sqrt[b^2*k^2 + a^2*l^2], t -> -ArcCos[(a*l)/Sqrt[b^2*k^2 + a^2*l^2]]}, {lambda -> (a*b)/Sqrt[b^2*k^2 + a^2*l^2], t -> ArcCos[(a*l)/Sqrt[b^2*k^2 + a^2*l^2]]}} The warning suggests that some solutions might be missing. In fact, Solve produces two extra expressions that are not solutions. eq /. sol // Simplify[#, {a > b > 0, k > 0, l\[Element]Reals}] & returns {False,True,True,False} My questions are: 1. Is that a bug or feature that Solve produces expressions that are not solutions for the original equations? 2. How to make Solve to return only 'true' solutions? On a side note, I tried Reduce only once because Mathematica froze my PC. AK

**Follow-Ups**:**Re: Re: Reduce/Solve***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**References**:**Re: Reduce/Solve***From:*"Dana DeLouis" <delouis@bellsouth.net>

**Re: Re: Reduce/Solve***From:*DrBob <drbob@bigfoot.com>