# Determine the ratio in which the straight line

Question:

Determine the ratio in which the straight line x − y − 2 = 0 divides the line segment joining (3, −1) and (8, 9).

Solution:

Let the line divide the line segment joining the points A (3,−1) and B (8, 9) in the ratio  at any point

Now according to the section formula if point a point P divides a line segment joining andin the ratio m: n internally than,

$\mathrm{P}(x, y)=\left(\frac{n x_{1}+m x_{2}}{m+n}, \frac{n y_{1}+m y_{2}}{m+n}\right)$

So,

$\mathrm{P}(x, y)=\left(\frac{8 \lambda+3}{\lambda+1}, \frac{9 \lambda-1}{\lambda+1}\right)$

Since, P lies on the given line. So,

$x-y-2=0$

Put the values of co-ordinates of point P in the equation of line to get,

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

On further simplification we get,

$-3 \lambda+2=0$

So, $\lambda=\frac{2}{3}$

So the line divides the line segment joining A and B in the ratio 2: 3 internally.