Determine the ratio in which the line 3x+y−9=0 divides the line segment joining the points (1,3) and (2,7).
Using the section formula, if a point (x,y) lying on the given line divides the line joining the points (x1,y1) and (x2,y2) in the ratio m:n, then (x,y)=(mx2+nx1m+n,my2+ny1m+n)
Let the ratio be k:1
Substituting (x1,y1)=(1,3) and (x2,y2)=(2,7) in the section formula, we get,
P = (k(2)+1(1)k+1,k(7)+1(3)k+1)=(2k+1k+1,7k+3k+1)
Since P lies on the line 3x+y−9=0, we have
3(2k+1k+1)+(7k+3k+1)−9=0
6k+3+7k+3−9(k+1)=0
13k+6−9k−9=0
4k−3=0
k=34
Hence, the ratio is 3:4