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MathGroup Archive 2004

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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


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