Question

Prove that the line y − x + 2 = 0 divides the join of points (3, −1) and (8, 9) in the ratio 2 : 3.

Answer 1

Let y − x + 2 = 0 divide the line joining the points (3, −1) and (8, 9) at point P in the ratio k : 1

$\therefore P\equiv \left(\frac{3+8k}{k+1},\frac{-1+9k}{k+1}\right)$

P lies on the line y − x + 2 = 0

$$\begin{gathered} \therefore \frac{-1+9k}{k+1}-\frac{3+8k}{k+1}+2=0 \\ \Rightarrow -1+9k-3-8k+2k+2=0 \\ \Rightarrow 3k=2 \\ \Rightarrow k=\frac{2}{3} \end{gathered}$$

Hence, the line y − x + 2 = 0 divides the line joining the points (3, −1) and (8, 9) in the ratio 2 : 3