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Performing the shape derivative (Sokolowski and Zolesio, 1992) and using the maximum principle, we show that the so-called Quadrature Surfaces free boundary problem egin{equation*} Q_S(f,k) left{ egin{array}{l} -Delta u_{Omega}=f quad ext{in  }Omega\ u_{Omega}=0ext{ on  }partial Omega\  left| abla u_{Omega }ight|=k ;(ext{constant})ext{  on }partial  Omega. end{array} ight. end{equation*} has a solution which contains strictly the support of $f$ if and only if $$int_Cf(x)dx>kint_{partial C}dsigma.$$  Where $C$ is the convex hull of the support of $f$. We also give  a necessary and sufficient condition of existence for the problem $Q_S(f,k)$ where the term source $f$ is a uniform density supported by a segment

Necessary and Sufficient Condition of Existence for the Quadrature Surfaces Free Boundary Problem