Question

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

Answer 1

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

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

${\displaystyle \Rightarrow \frac{k\left(2\right)+\frac{1}{2}}{k+1}=\frac{3}{4} \text{and} \frac{k(-5)+\frac{3}{2}}{k+1}=\frac{5}{12}}$

${\displaystyle \Rightarrow 8k+2=3k+3\text{and}-60k+18=5k+5}$

${\displaystyle \Rightarrow k=\frac{1}{5}\text{and}k=\frac{1}{5}}$
Hence, the required ratio is1:5