 Geometry

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 point $R$ (the sought-after external division point) is on the line $x+y = -18,$ then what is the positive constant $k?$

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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