Question

Find 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 $\left(2,-5\right)$. [CBSE 2015]

Answer 1

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 $\left(2,-5\right)$. Then
$$\begin{gathered} \left(\frac{3}{4},\frac{5}{12}\right)=\left(\frac{k\left(2\right)+{\displaystyle \frac{1}{2}}}{k+1},\frac{k\left(-5\right)+{\displaystyle \frac{3}{2}}}{k+1}\right) \\ \Rightarrow \frac{k\left(2\right)+{\displaystyle \frac{1}{2}}}{k+1}=\frac{3}{4}\mathrm{and}\frac{k\left(-5\right)+{\displaystyle \frac{3}{2}}}{k+1}=\frac{5}{12} \\ \Rightarrow 8k+2=3k+3\mathrm{and}-60k+18=5k+5 \\ \Rightarrow k=\frac{1}{5}\mathrm{and}k=\frac{1}{5} \end{gathered}$$
Hence, the required ratio is 1 : 5.