Question

The ratio in which the point $P\left(\frac{3}{4},\frac{5}{12}\right)$ divides the line segment joining the points A(1/2, 3/2) and B(2, –5) is _________.

Answer 1

Section formula: if the point (x, y) divides the line segment joining the points (x₁, y₁) and (x₂, y₂) internally in the ratio k : 1, then the coordinates (x, y) = $\left(\frac{k{x}_2+x_1}{k+1},\frac{k{y}_2+y_1}{k+1}\right)$

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

Therefore, using section formula, the coordinates of P are:

$$\begin{gathered} \left(\frac{3}{4},\frac{5}{12}\right)=\left(\frac{k\left(2\right)+1\left({\displaystyle \frac{1}{2}}\right)}{k+1},\frac{k\left(-5\right)+1\left({\displaystyle \frac{3}{2}}\right)}{k+1}\right) \\ \Rightarrow \left(\frac{3}{4},\frac{5}{12}\right)=\left(\frac{2k+{\displaystyle \frac{1}{2}}}{k+1},\frac{-5k+{\displaystyle \frac{3}{2}}}{k+1}\right) \\ \Rightarrow \left(\frac{3}{4},\frac{5}{12}\right)=\left(\frac{{\displaystyle \frac{4k+1}{2}}}{k+1},\frac{{\displaystyle \frac{-10k+3}{2}}}{k+1}\right) \\ \Rightarrow \left(\frac{3}{4},\frac{5}{12}\right)=\left(\frac{{\displaystyle 4k+1}}{2k+2},\frac{{\displaystyle -10k+3}}{2k+2}\right) \\ \Rightarrow \frac{{\displaystyle 4k+1}}{2k+2}=\frac{3}{4} \\ \Rightarrow 4\left(4k+1\right)=3\left(2k+2\right) \\ \Rightarrow 16k+4=6k+6 \\ \Rightarrow 10k=2 \\ \Rightarrow k=\frac{2}{10} \\ \Rightarrow k=\frac{1}{5} \end{gathered}$$

Hence, the ratio in which the point $P\left(\frac{3}{4},\frac{5}{12}\right)$ divides the line segment joining the points A(1/2, 3/2) and B(2, –5) is 1 : 5.