2D Coordinate Geometry

Section Formula


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

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

Let PP be the internal division point of AB\overline{AB} joining points AA and BB in the ratio 7:8,7:8, and QQ be the external division point of AB\overline{AB} in the ratio 8:7.8:7. If the length of PQ\overline{PQ} is PQ=113,\lvert\overline{PQ}\rvert=113, what is AB?\lvert\overline{AB}\rvert?

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

The line segment ABAB connecting two points A=(23,14)A=(23, 14) and B=(2,5)B=(2, 5) on the xy-plane is internally divided 2:12:1 by a point P=(x,yP=(x, y). What is x+yx+y?


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