       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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