Consider the following system of equations: ( x + 1 ) 2 [ ( p − l ) 2 + ( q − m ) 2 ] = ( a − l ) 2 + ( b − m ) 2 ( x + 1 ) 2 [ ( p − a ) 2 + ( q − b ) 2 ] = x 2 [ ( a − l ) 2 + ( b − m ) 2 ] Find the value of p and q in terms of a , b , x , l and m, provided that x &gt; 0.

Question

Consider the following system of equations:                    (        x        +        1                  )          2                [        (        p        −        l                  )          2                +        (        q        −        m                  )          2                ]                            =        (        a        −        l                  )          2                +        (        b        −        m                  )          2                                    (        x        +        1                  )          2                [        (        p        −        a                  )          2                +        (        q        −        b                  )          2                ]                            =                  x          2                [        (        a        −        l                  )          2                +        (        b        −        m                  )          2                ]            Find the value of   p and   q in terms of   a  ,  b  ,  x  ,  l and   m, provided that   x  &amp;gt;  0.

rcampas4i · Accepted Answer

Divide both equations by   (  x  +  1      )    2   and set      α    2    =            (      a      −      l              )        2            +      (      b      −      m              )        2                    (      x      +      1              )        2                    β    2    =                    x        2            [      (      a      −      l              )        2            +      (      b      −      m              )        2            ]              (      x      +      1              )        2              ,then the system of equations to solve becomes      {                            (          p          −          l                      )            2                    +          (          q          −          m                      )            2                                    =                                      α            2                                                (          p          −          a                      )            2                    +          (          q          −          b                      )            2                                    =                                      β            2                                  you have to find the intersection of two circles in the plane. One of these circle has center       c    1    =  (  l  ,  m  ) and radius       r    1    =  α while the other has center       c    2    =  (  a  ,  b  ) and radius       r    2    =  β.

Consider the following system of equations: (x+1)^2[(p−l)^2+(q−m)^2] x+1)^2[(p−a)^2+(q−b)^2]=(a−l)^2+(b−m)^2=x^2[(a−l)^2+(b−m)^2] Find the value of p and q in terms of a,b,x,l and m, provided that x>0.

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