Determine the ratio, in which the line $2x + y - 4 = 0$ divides the line segment joining the points $A(2, -2)$ and $B(3, 7)$.

Given:

The line $2x + y - 4 = 0$ divides the line segment joining the points $A(2, -2)$ and $B(3, 7)$.

To do:

We have to find the ratio of division.

Solution:

Let the straight line $2x + y - 4 = 0$ divides the line segment joining the points $A(2, -2), B(3, 7)$ in the ratio $m : n$ at point $(x_1,y_1)$.

Using the section formula, if a point $( x,\ y)$ divides the line joining the points $( x_1,\ y_1)$ and $( x_2,\ y_2)$ in the ratio $m:n$, then 

$(x,\ y)=( \frac{mx_2+nx_1}{m+n},\ \frac{my_2+ny_1}{m+n})$

Therefore,

$(x_1,y_1)=(\frac{m \times 3+n \times 2}{m+n}, \frac{m \times 7+n \times (-2)}{m+n})$

$=(\frac{3m+2n}{m+n}, \frac{7m-2n}{m+n})$

The point $(x, y)$ lies on the line $2x+y-4=0$.

This implies, the point $(x,y)$ satisfies the above equation.

$\Rightarrow 2(\frac{3m+2n}{m+n})+\frac{7m-2n}{m+n}-4=0$

$\Rightarrow(6m+4n)+(7m-2n)-4(m+n)=0$

$\Rightarrow 6m+4n+7m-2n-4m-4n=0$

$\Rightarrow 9m-2n=0$

$\Rightarrow 9m=2n$

$\Rightarrow \frac{m}{n}=\frac{2}{9}$

$\Rightarrow m:n=2:9$

The required ratio of division is $2:9$. 

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Updated on: 10-Oct-2022

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