Find the ratio in which the points $P (\frac{3}{4} , \frac{5}{12})$ divides the line segments joining the points $A (\frac{1}{2}, \frac{3}{2})$ and $B (2, -5)$.

Given:

Points $P (\frac{3}{4} , \frac{5}{12})$ divides the line segments joining the points $A (\frac{1}{2}, \frac{3}{2})$ and $B (2, -5)$.

To do:

We have to find the ratio of division.

Solution:

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 

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

Here,

$x_1=\frac{1}{2}, y_1=\frac{3}{2}, x_2=2, y_2=-5$

Let the ratio be $m:n$

This implies,

$P (\frac{3}{4} , \frac{5}{12})=( \frac{m(2)+n(\frac{1}{2})}{m+n},\ \frac{m(-5)+n(\frac{3}{2})}{m+n})$

Therefore, equating coordinates on both sides, we get,

$\frac{3}{4}=\frac{m(2)+n(\frac{1}{2})}{m+n}$

$\Rightarrow  3(m+n)=4(2m+\frac{1}{2}n)$

$\Rightarrow  3m+3n=8m+2n$

$\Rightarrow  8m-3m=3n-2n$

$\Rightarrow  5m=n$

$\Rightarrow  \frac{m}{n}=\frac{1}{5}$

​$\Rightarrow  m:n=1:5$ 

 The required ratio is $1:5$.