Find the ratio in which the line segment joining the points A(3, −3) and B(−2, 7) is divided by the *x*-axis. Also, find the coordinates of the point of division.

#### Solution

Suppose the *x*-axis divides the line segment joining the points A(3, −3) and B(−2, 7) in the ratio *k* : 1.

Using section formula, we get

Coordinates of the point of division = \[\left( \frac{- 2k + 3}{k + 1}, \frac{7k - 3}{k + 1} \right)\]

Since the point of division lies on the *x*-axis, so its *y*-coordinate is 0.

\[\therefore \frac{7k - 3}{k + 1} = 0\]

\[ \Rightarrow 7k - 3 = 0\]

\[ \Rightarrow k = \frac{3}{7}\]

So, the required ratio is \[\frac{3}{7}\] : 1 or 3 : 7.

Putting *k* = \[\frac{3}{7}\] , we get

Coordinates of the point of division = \[\left( \frac{- 2 \times \frac{3}{7} + 3}{\frac{3}{7} + 1}, 0 \right) = \left( \frac{- 6 + 21}{3 + 7}, 0 \right) = \left( \frac{15}{10}, 0 \right) = \left( \frac{3}{2}, 0 \right)\]