Question

Consider a line l with equation ax + by + c = 0. Show that the expression ax₁ + by₁ +c is positive for (x₁, y₁) on one side of the line l and negative for all points on the other side.

Answer 1

Note that a point (x₁, y₁) lies on the line l if and only if ax₁ +by₁ +c = 0.

Suppose that points (α, β) and (γ, δ) are on the same side of l. Then the line segment from (α, β) and (γ, δ) does not cross l. So ax₁ + by₁ + c≠ 0 for each point (x₁, y₁) on the line segment. Thus the expression ax₁ + by₁ + c cannot change sign as the point (x₁, y₁) moves along the line segment from (α, β) to (γ, δ). Hence the signs of aα + bβ + c and aγ + bδ + c are the same. 

Points on different sides of l must have different signs, as it is possible to choose (x₁, y₁) so that ax₁ + by₁ + c > 0, and choose (x₂, y₂) so that ax₂ + by₂ + c < 0.