The equation of line joining #(x_1,y_1)# and #(x_2,y_2)# is #(y-y_1)/(y_2-y_1)=(x-x_1)/(x_2-x_1)#
Hence equation of line joining #(1,3)# and #(2,7)# is
#(y-3)/(7-3)=(x-1)/(2-1)# or #(y-3)/4=(x-1)/1#
i.e. #4x-4=y-3# or #y=4x-1#
Solution of equations #3x+y=9# and #y=4x-1# gives point of intersection. Putting second equation in first we get
#3x+4x-1=9# or #x=10/7# and #y=4xx10/7-1=33/7#
i.e. point of intersection is #(10/7,33/7)#
Now distance of #(10/7,33/7)# and #(1,3)# is
#sqrt((10/7-1)^2+(33/7-3)^2)=sqrt(9/49+144/49)=sqrt153/7#
and distance of #(10/7,33/7)# and #(2,7)# is
#sqrt((10/7-2)^2+(33/7-7)^2)=sqrt(16/49+256/49)=sqrt272/7#
and ratio is #sqrt153/sqrt272=sqrt(17xx3xx3)/sqrt(17xx4xx4)=3/4#
Hence, the line joining the points #(1,3)# and #(2,7)# is divided by the line #3x+y=9# in the ratio of #3:4#.
graph{(3x+y-9)(y-4x+1)((x-1)^2+(y-3)^2-0.01)((x-2)^2+(y-7)^2-0.01)=0 [-2.96, 7.04, 2.5, 7.5]}
