find in what ratio is the line joining (3,4) and (5,-7) cut by the co-ordinates axis     

Expert Answers
lfryerda eNotes educator| Certified Educator

To find the ratio of the line cut by the x-axis, we need to use the distance formula `d=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}` between the two points `(x_1,y_1)` and `(x_2,y_2)`.

The equation of the line between the end points is found by getting the slope of the line.



The equation of the line is then found using `y=mx+b` and solving for b.

`4=-11/2\cdot 3+b`



so the equation of the line is `y=-11/2 x +41/2` .

The x-intercept of this line is at `y=0`. That is, when `0=-11/2 x+41/2`

which is at `x=41/11`.

Consider the three points `A=(3,4)`, `B=(41/11,0)` and `C=(5,-7)`. The point B is the x-intercept of the line. 

The distance from A to B is




and the distance from B to C is




This means that the ratio from AB to BC after cancelling out the common factor of 11 is




jeew-m eNotes educator| Certified Educator

The general equation for a line is y = mx+c

Our line goes through (3,4) and (5,-7)

So we can write;




11 = -2m

  m= -11/2

From (1) we can get 4=(-11/2)*3+c

                              c =41/2

Equation of the line joining (3,4) and (5,-7) is y=(-11/2)x+41/2


Since y values of two points changes 4 to -7 the line will cut x axis.

When y=0 then x=(41/2)(2/11) = 41/11


So the cutting point is (41/11,0)

Let legth AB=length (3,4) to ((41/11,0)

             BC= length (5,-7) to ((41/11,0)


AB=sqrt[(3-41/11)^2+(4-0)^2] = sqrt[2000/121]

BC=sqrt[(5-41/11)^2+(-7-0)^2]= sqrt[6125/121]


So the ratio is AB/BC  =sqrt[2000/121]/sqrt[6125/121]


                                = 4/7


The the cordinate axis cut the line to the ratio of 4:7