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

lfryerda | 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.

m={-7-4}/{5-3}

=-11/2

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

4=-11/2\cdot 3+b

b={8+33}/2

b=41/2

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

d_{AB}=\sqrt{(3-41/11)^2+(0-4)^2}

=\sqrt{64/121+16}

=\sqrt{2000}/11

and the distance from B to C is

d_{BC}=\sqrt{(5-41/11)^2+(-7-0)^2}

=\sqrt{196/121+49}

=\sqrt{6125}/11

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

\sqrt{2000/6125}

=\sqrt{16/49}

=4/7

jeew-m | 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;

4=m*3+c------------(1)

-7=m*5+c------------(2)

(1)-(2)

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]

=sqrt[2000/6125]

= 4/7

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