Question #268659

If A is the point (−3, 5, 10) and B is the point (12, −5, −15), find the coordinates of point P on the line AB given that AP: PB = 2: 3.


1
Expert's answer
2021-11-22T15:50:11-0500

Using the section formula, if a point (x,y,z) divides the line joining the points (x1,y1,z1)(x_1, y_1, z_1) and (x2,y2,z3)(x_2, y_2, z_3) in the ratio m:nm:n, then;

(x,y,z)=(mx2+nx1m+n,my2+ny1m+n,mz2+nz1m+n)AP:PB=2:3m:n=2:3(x,y,z)=(2(12)+3(3)2+3,2(5)+3(5)3+2,2(15)+3(10)3+2)(x,y,z)=(2495,10+155,30+305)(x,y,z)=(3,1,0)P    (x,y)=(3,1)(x,y,z) = (\dfrac{mx_2+ nx_1}{m+n}, \dfrac{my_2+ ny_1}{m+n}, \dfrac{mz_2+ nz_1}{m+n})\\ \quad\\ AP: PB = 2: 3\\ m: n = 2:3\\ \quad\\ (x,y,z) = (\dfrac{2(12)+ 3(-3)}{2+3}, \dfrac{2(-5)+ 3(5)}{3+2}, \dfrac{2(-15)+ 3(10)}{3+2})\\ \quad\\ (x,y,z) = (\dfrac{24-9}{5}, \dfrac{-10+15}5, \dfrac{-30+30} 5)\\ (x,y,z) = ( 3,1,0)\\ \quad\\ P \implies (x,y) = (3,1)

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