Find the ratios in which the following straight lines divide the line segments joining the given points. State whether the points lie on the same side or on either side of the straight line.
$2 x + 3 y = 5$ ; $(0, 0)$ and $(- 2, 1)$ .

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Solution

According to question,
$s t r a i g h t l i n e : 2 x + 3 y - 5 = 0 a n d, j o i n i n g p o i n t s (0, 0) & (- 2, 1) l e t t h e r a t i o b e m : n = k : 1, H e r e, A (x_{1}, y_{1}), B (x_{2} y_{2}) [c o - o r d i n a t e o f c f r o m s e c t i o n f o r m u l a,] c = (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n}) = (\frac{k (- 2) + 1 (0)}{k + 1}, \frac{k (1) + 1 (0)}{k + 1}) c = \frac{- 2 k}{k + 1}, \frac{k}{k + 1} s o, c s a t i s f y t h e e q u a t i o n o f l i n e : \Rightarrow 2 x + 3 y - 5 = 0 \Rightarrow 2 (\frac{- 2 k}{k + 1}) + 3 (\frac{k}{k + 1}) - 5 = 0 \Rightarrow \frac{- 4 k + 3 k - 5 (k + 1)}{k + 1} = 0 \Rightarrow \frac{- k - 5 k - 5}{k + 1} = 0 ∴ k = \frac{- 5}{6} (e x t e r n a l d i v i s i o n) s o, t h a t r a t i o i s 5 : 6 (e x t e r n a l l y) a n d t h i s i s c o r r e c t r a t i o .$

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