If $P\left(x,y\right)$  lies on the line joining $A\left(x_1,y_1\right)$  and $B\left(x_2,y_2\right)$ , then $P$  divides $\overline{AB}$  in the ratio $x_1-x:x-x_2$  or $y_1-y:y-y_2$ .

X-axis divides the line segment joining $\left(x_1,y_1\right)$  and $\left(x_2,y_2\right)$  in the ratio $-y_1:y_2$ .

Y-axis divides the line segment joining $\left(x_1,y_1\right)$  and $\left(x_2,y_2\right)$  in the ratio $-x_1:x_2$

Line Division and Coordinate Geometry Principles

Point Division on a Line Segment

If a point P(x,y) lies on the line joining points A(x₁,y₁) and B(x₂,y₂), then point P divides the line segment AB in the ratio:

x₁ - x : x - x₂ or y₁ - y : y - y₂

Division by the X-axis

The X-axis divides the line segment joining points (x₁,y₁) and (x₂,y₂) in the ratio:

-y₁ : y₂

This occurs because the Y-coordinate on the X-axis is zero.

Division by the Y-axis

The Y-axis divides the line segment joining points (x₁,y₁) and (x₂,y₂) in the ratio:

-x₁ : x₂

This occurs because the X-coordinate on the Y-axis is zero.

Application

These ratios are fundamental in coordinate geometry and are used to find points of intersection of line segments with coordinate axes using the section formula. They help determine where a line segment crosses the X-axis or Y-axis and in what proportion it divides the original segment.