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

\[\therefore \frac{- 1 + 9k}{k + 1} - \frac{3 + 8k}{k + 1} + 2 = 0\]

\[ \Rightarrow - 1 + 9k - 3 - 8k + 2k + 2 = 0\]

\[ \Rightarrow 3k = 2\]

\[ \Rightarrow k = \frac{2}{3}\]

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