Position of a Point With Respect to a Line

Summary

This math recipe will help you find the position of a point with respect to a line in the Cartesian plane.

Skill Level

Easy

Time

Approx. 1 min

Ingredients

Equation of the line: ax + by + c = 0

Coordinates of the points: (x₁, y₁) and (x₂, y₂)

Method

To find the position of the points with respect to the line, we’ll substitute the coordinates of the points in the equation of the line, to get the following expressions:

ax₁ + by₁ + c

ax₂ + by₂ + c

If the two expressions have the

same sign, then the points lie on the same side of the line.

opposite sign, then the points lie on the opposite side of the line.

Examples

Example 1 Find the position of the points (1, 3) and (0, –4) with respect to the line 3x + 2y – 4 = 0.

Solution To find the relative positions of the points, we’ll substitute their coordinates in the equation of the line separately.

On substituting (1, 3), we’ll get

3(1) + 2(3) – 4 = 5

On substituting (0, –4), we’ll get

3(0) + 2(–4) – 4 = –12

Since the two expressions are of the opposite sign, the two points lie on the opposite side of the given line. The figure shows the line and the two points.

Example 2 Find the position of the points (–2, 3) and (1, 4) with respect to the line x – 4y + 3 = 0.

Solution To find the relative positions of the points, we’ll substitute their coordinates in the line’s equation separately.

On substituting (–2, 3), we’ll get

(–2) – 4(3) + 3 = –11

On substituting (1, 4), we’ll get

1 – 4(4) + 3 = –12

Since the two expressions are of the same sign, the two points lie on the same side of the given line. The figure shows the line and the two points.

That’s it for this recipe. Hope you found it helpful.

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