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MathGroup Archive 1993

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Eliminate

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: Eliminate
  • From: puglisi at settimo.italtel.it
  • Date: Mon, 4 Oct 93 15:43:10 MDT

>>> Sergio Rescia writes:
> 
> I am trying to use Eliminate to eliminate the unknown y in a system of 2 equations:
> 
> In[1]:= Eliminate[{ (x-a)^2+(y-b)^2==r^2 , 
>                     x^2+y^2==9 },            {y}]
> 
>            4    2           2      2
> Out[1]= r  + r  (-18 - 2 a  - 2 b  + 4 a x) == 
>  
>                2    4       2      2  2    4               3          2
> >    -81 - 18 a  - a  + 18 b  - 2 a  b  - b  + 36 a x + 4 a  x + 4 a b  x - 
>  
>          2  2      2  2
> >     4 a  x  - 4 b  x
> 
> 
> Is there any way to force Eliminate to produce a result in the form:
> 
> 	expr==0
>
> Or better yet there is any smart way to compute directly the discriminant of the second degree equation 
> (in x) Out[1]?
> 
> Thank you.
> 
> Sergio rescia
> 
>
 
Here is an answer:

In[5]:= eq={(-a + x)^2 + (-b + y)^2 == r^2, x^2 + y^2 == 9}

Out[5]= {(-a + x)^2 + (-b + y)^2 == r^2, x^2 + y^2 == 9}

In[6]:= Solve[eq,x,y]

Out[6]= {{x -> (-(a*(-4 - 36/(a^2 + b^2) + (4*r^2)/(a^2 + b^2))) + 
        (a^2*(-4 - 36/(a^2 + b^2) + (4*r^2)/(a^2 + b^2))^2 - 
           16*(18 + a^2 + b^2 + 81/(a^2 + b^2) - (36*b^2)/(a^2 + b^2) - 
              2*r^2 - (18*r^2)/(a^2 + b^2) + r^4/(a^2 + b^2)))^(1/2))/8}, 
   {x -> (-(a*(-4 - 36/(a^2 + b^2) + (4*r^2)/(a^2 + b^2))) - 
        (a^2*(-4 - 36/(a^2 + b^2) + (4*r^2)/(a^2 + b^2))^2 - 
           16*(18 + a^2 + b^2 + 81/(a^2 + b^2) - (36*b^2)/(a^2 + b^2) - 
              2*r^2 - (18*r^2)/(a^2 + b^2) + r^4/(a^2 + b^2)))^(1/2))/8}}

In[7]:= Eliminate[eq,y]

Out[7]= b != 0 && a^4 - 4*a^3*x + a*(-36 - 4*b^2 + 4*r^2)*x + 
      a^2*(18 + 2*b^2 - 2*r^2 + 4*x^2) == 
     -81 + 18*b^2 - b^4 + 18*r^2 + 2*b^2*r^2 - r^4 - 4*b^2*x^2 || 
   b == 0 && a^2 - 2*a*x == -9 + r^2




Alberto Puglisi
Italtel R&D
Milan, Italy		Internet:   puglisi at settimo.italtel.it





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