In what ratio does the line x − y − 2 = 0 divide the line segment joining the points
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)?

Solution:

Let the line x − y − 2 = 0 divide the line segment joining the points A(3, −1) and B(8, 9) in the ratio : 1 at P.
Then, the coordinates of P are

$P\left(\frac{8 k+3}{k+1}, \frac{9 k-1}{k+1}\right)$

Since, P lies on the line x − y − 2 = 0, we have:

$\left(\frac{8 k+3}{k+1}\right)-\left(\frac{9 k-1}{k+1}\right)-2=0$

$\Rightarrow 8 k+3-9 k+1-2 k-2=0$

$\Rightarrow 8 k-9 k-2 k+3+1-2=0$

$\Rightarrow-3 k+2=0$

$\Rightarrow-3 k=-2$

$\Rightarrow k=\frac{2}{3}$

So, the required ratio is $\frac{2}{3}: 1$, which is equal to $2: 3$.