In what ratio is the line joining the points (1,3) and (2,7) divided by the line 3x+y=9?

1 Answer
Sep 8, 2017

The line joining the points (1,3) and (2,7) is divided by the line 3x+y=9 in the ratio of 3:4.

Explanation:

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]}