Let \(P\) be the internal division point of the line segment \(AB\) joining points \(A=(-9,6)\) and \(B=(12,4)\) in the ratio \(t:(1-t)\) for \(t>0\). If \(P\) lies in the first quadrant of the Cartesian plane, what is the range of the real number \(t?\)

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Consider the line segment joining points \(A=(-6,-7)\) and \(B=(8,-1)\). If \(P\) is the intersection point of this line segment and the \(y\text{-axis},\) and \(P\) internally divides the line segment in the ratio \(m:n,\) where \(m\) and \(n\) are coprime positive integers, what is the value of \(mn?\)

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Let \(P\) be the internal division point of \(\overline{AB}\) joining points \(A\) and \(B\) in the ratio \(7:8,\) and \(Q\) be the external division point of \(\overline{AB}\) in the ratio \(8:7.\) If the length of \(\overline{PQ}\) is \(\lvert\overline{PQ}\rvert=113,\) what is \(\lvert\overline{AB}\rvert?\)

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Let \(R\) be the external division point of the line segment \(PQ\) joining points \(P=(-1,1)\) and \(Q=(11,16)\) in the ratio \(k:5.\) If point \(R\) (the sought-after external division point) is on the line \(x+y = -18,\) then what is the positive constant \(k?\)

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The line segment \(AB\) connecting two points \(A=(23, 14)\) and \(B=(2, 5)\) on the xy-plane is internally divided \(2:1\) by a point \(P=(x, y\)). What is \(x+y\)?

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