In what ratio is the line joining the points 2 3 and 4?

Text Solution

Solution : The equation of the line joining the points `(6, 8)` and `(-3, -2)` is `y − 8 = frac{− 2 − 8}{ − 3 − 6} ( x − 6 )` <br>`⇒ 10 x − 9 y + 12 = 0` <br> Let `10x - 9y + 12 = 0` divide the line joining the points `(2, 3)` and `(4, -5)` at point `P` in the ratio `k : 1` `∴ P = ( frac{4 k + 2} {k + 1} , frac{− 5 k + 3}{ k + 1 })` <br> `P` lies on the line `10x - 9y + 12 = 0` `∴ 10 ( frac{4 k + 2}{ k + 1} ) − 9 (frac{ − 5 k + 3}{ k + 1} ) + 12 = 0` <br>`⇒ 40 k + 20 + 45 k − 27 + 12 k + 12 = 0` <br>`⇒ 97 k + 5 = 0` <br> `⇒ k =frac{ − 5}{ 97}`

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.