Mutual position of two straight lines -2

author: Петя
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Българска версия:

  Let's assume that two straight lines g₁ and g₂ are defined by the equations: g₁: A₁x+ B₁y+ C₁ = 0, g₂:A₂x+ B₂y + C₂ = 0.

    We know that the two straight lines fuse together when the coefficients in their equations are proportionately, i.e. when a number v exists, that (1) А₂ = vA₁, B₂ = vB₁, C₂ = vB₁.

            When the condition (1) is present, the straight lines g₁ and g₂ are intersecting or parallel. Which one of the two cases is available can be understood by solving the system (2)

                        A₁x + B₁y + C₁ = 0

                        A₂x + B₂y + C₂ = 0

        If the system (2) has one solution (х₁; у₁), the point М₁ (х₁; у₁) satisfies the equations of the two straight lines g₁ and g₂, i.е g₁ and g₂ are intesected in point М₁.

            If the system (2) has no solution the straight lines g₁ and g₂  don't have a common point, i.е. they are parallel.

        The mutual position of g₁ and g₂ can be defined also without solving the system (2).

            We know that the vectors u₁ ^→(-B₁; A₁) and u₂ ^→(-B₂;A₂)  are collinear respectively to the straight lines g₁ and g₂. The straight lines are parallel  if and only if the vectors u₁ ^→and u₂^→ are collinear, i.е. when a number v exist that u₂ ^→ = vu₁ ^→. Then А₂ = vA₁, B₂ = vB₁.

Consequently the straight lines g₁ and g₂:

• are parallel if А₂ = vA₁, B₂ = vB₁, but С is different from vC₁, i.е. the coefficient in front of х and у in their equations are proportionately, but this proportionality isn't standed by the free terms;

• are collineral if the coefficient before х and у in their equations aren't proportionately (А₁/А₂  is  different from В₁/В₂).

     In case of the two straight lines have Cartesian equations g₁: y = k₁x+ n₁, g₂:y = k₂x + n₂, having in mind geometrical meaning of the angular coefficient and the segment of the от axis Оу, are the following conditions emerge about the mutual different positions of the two straight lines:

1.          k₁ different from k₂ - g₁ and g₂ are collineral;

2.          k₁ = k₂, n₁ different from n₂ - g₁ and g₂ are parallel;

3.          k₁ = k₂, n₁ = n₂ - g₁ and g₂ are merged.