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Find the ratio in which the point \( \mathrm{P}\left(\frac{3}{4}, \frac{5}{12}\right) \) divides the line segment 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 $1:5$.