Find the ratio in which the point

Let $k: 1$ be the ratio in which the point $P\left(\frac{3}{4}, \frac{5}{12}\right)$ divides the line segment joining the points $A\left(\frac{1}{2}, \frac{3}{2}\right)$ and $(2,-5)$. Then

$\left(\frac{3}{4}, \frac{5}{12}\right)=\left(\frac{k(2)+\frac{1}{2}}{k+1}, \frac{k(-5)+\frac{3}{2}}{k+1}\right)$

$\Rightarrow \frac{k(2)+\frac{1}{2}}{k+1}=\frac{3}{4} \quad$ and $\frac{k(-5)+\frac{3}{2}}{k+1}=\frac{5}{12}$

$\Rightarrow 8 k+2=3 k+3$ and $-60 k+18=5 k+5$

$\Rightarrow k=\frac{1}{5} \quad$ and $k=\frac{1}{5}$

Hence, the required ratio is 1 : 5.