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