Question

In what ratio does the line x − y − 2 = 0 divide the line segment joining the points A(3, −1) and B(8, 9)?

Answer 1

Let the line x − y − 2 = 0 divide the line segment joining the points A(3, −1) and B(8, 9) in the ratio k : 1 at P.
Then, the coordinates of P are
$P\left(\frac{8k+3}{k+1},\frac{9k-1}{k+1}\right)$
Since, P lies on the line x − y − 2 = 0, we have:

$$\begin{gathered} \left(\frac{8k+3}{k+1}\right)-\left(\frac{9k-1}{k+1}\right)-2=0 \\ \Rightarrow 8k+3-9k+1-2k-2=0 \\ \Rightarrow 8k-9k-2k+3+1-2=0 \\ \Rightarrow -3k+2=0 \\ \Rightarrow -3k=-2 \\ \Rightarrow k=\frac{2}{3} \end{gathered}$$
So, the required ratio is $\frac{2}{3}$ : 1, which is equal to 2 : 3.