Question

Find the ratio in which the line joining the points (2, 4, 16) and (3, 5, -4) is divided by the plane 2x – 3y+ z+ 6 = 0. Also find the co-ordinates of the point of division

Answer 1

Explanation:

let the plane intersects the join of points in the ratio k:1

let (x,y,z) be the point of intersection.

`x=(3k+2)/(k+1) \\y=(5k+4)/(k+1) \\z=(-4k+16)/(k+1) \\\because ~(x,y,z)~lies~on~the~plane.\\2((3k+2)/(k+1) )-3((5k+4)/(k+1) )+(-4k+16)/(k+1) +6=0\\multiply~by~k+1\\2(3k+2)-3(5k+4)+(-4k+16)+6(k+1)=0\\6k+4-15k-12-4k+16+6k+6=0\\-7k+14=0\\k=2\\x=(3*2+2)/(2+1) =(8)/(3) \\y=(5*2+4)/(2+1)= (14)/(3) \\z=(-4*2+16)/(2+1) =(8)/(3)`

point of intersection is (8/3,14/3,8/3)

and ratio of division is 2:1