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).

Answer 1

The equation of the line joining the points (6, 8) and (−3, −2) is

$$\begin{gathered} y-8=\frac{-2-8}{-3-6}\left(x-6\right) \\ \Rightarrow 10x-9y+12=0 \end{gathered}$$

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

$$\begin{gathered} \therefore 10\left(\frac{4k+2}{k+1}\right)-9\left(\frac{-5k+3}{k+1}\right)+12=0 \\ \Rightarrow 40k+20+45k-27+12k+12=0 \\ \Rightarrow 97k+5=0 \\ \Rightarrow k=-\frac{5}{97} \end{gathered}$$

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.