Given \(A=(5,0)\) and \(B=(0,4)\), what is the equation of the locus of points \(P\) such that \[ \lvert \overline{PA} \rvert^2- \lvert \overline{PB} \rvert^2=9?\]

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Given \(A=(9,-1)\) and \(B=(-9,5)\), what is the equation of the locus of points \(P\) such that \[ \lvert \overline{AP} \rvert: \lvert \overline{BP} \rvert=1:2 ,\] where \(\lvert \overline{AP} \rvert\) denotes the length of \(\overline{AP}?\)

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Let \(A=(6,-5)\) and \(B=(-6,1)\) be two vertices of triangle \( ABC.\) What is the locus of vertex \(C \) such that the centroid of \(\triangle ABC\) moves along the line \(2x+5y=1?\)

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Let \(A=(-6,0)\), \(B=(3,-3)\) and \(C=(5,3)\). What is the equation of the locus of points \(P\) such that \[\lvert \overline{AP} \rvert^2+\lvert \overline{BP} \rvert^2=2 \lvert \overline{CP} \rvert^2?\]

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If \(A=(-3,0)\) and \(B=(12,0),\) what is the equation for the locus of points \(P\) such that \(\lvert \overline{PA} \rvert =\lvert \overline{PB} \rvert,\) where \(\lvert \overline{PA} \rvert\) denotes the length of \( \overline{PA}?\)

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