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2D Coordinate Geometry

In the 1600s, René Descartes married algebra and geometry to create the Cartesian plane.

Section Formula

         

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 \(t?\)

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 \(mn?\)

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 \(\lvert\overline{AB}\rvert?\)

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 \(P\) is on the line \(x+y=-18,\) what is the positive number \(k?\)

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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