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Question
In what ratio is the line joining the points (2, 3) and (4, −5) divided by the line passing through the points (6, 8) and (−3, −2).
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Solution
The equation of the line joining the points (6, 8) and (−3, −2) is
\[y - 8 = \frac{- 2 - 8}{- 3 - 6}\left( x - 6 \right)\]
\[ \Rightarrow 10x - 9y + 12 = 0\]
Let 10x − 9y + 12 = 0 divide the line joining the points (2, 3) and (4, −5) at point P in the ratio k : 1
\[\therefore P \equiv \left( \frac{4k + 2}{k + 1}, \frac{- 5k + 3}{k + 1} \right)\]
P lies on the line 10x − 9y + 12 = 0
\[\therefore 10\left( \frac{4k + 2}{k + 1} \right) - 9\left( \frac{- 5k + 3}{k + 1} \right) + 12 = 0\]
\[ \Rightarrow 40k + 20 + 45k - 27 + 12k + 12 = 0\]
\[ \Rightarrow 97k + 5 = 0\]
\[ \Rightarrow k = - \frac{5}{97}\]
Hence, the line joining the points (2, 3) and (4, −5) is divided by the line passing through the points (6, 8) and (−3, −2) in the ratio 5 : 97 externally.
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