1. General type of equation of a line
The equation of type: Ax + By + C = 0 is called a general type equation of a line.
1. If A = O, we get the equation By + C = 0, ie y = – C / B, which is a line parallel to the x-axis, at a distance – C / B from the coordinate origin.
2. If B = O, the equation is reduced to Ax + C = 0, ie x = – C / A which is a line parallel to the y-axis, at a distance – C / A from the coordinate origin.
3. If B = 0, C = 0, the line equation is reduced to the form Ax = 0, ie x = 0. It coincides with the y-axis and represents its equation.
4. If A = 0, C = 0, the equation of a line is reduced to the form By = 0, ie y = 0. The line coincides with the x-axis and represents its equation.
2. Explicit kind of equation of a line
The equation у = kx + b, where k = tg а, is called an equation of a line in explicit form (fig. 4).
The number k is called the coefficient of the direction of the line, and the b-segment that the line intersects on the u-axis.
3. Segment type equation of a line
The equation of a line in segmental form is:
in which a and b are segments that intersect the lines on the coordinate axes x and y, respectively (fig. 5).
4. Normal type equation of a line
The normal type of equation of a line given in general is
in which the sign before the root is opposite to the sign before the coefficient C.
5. Equation of lines through one and two points
The equation of a line that passes through a given point M (x₁, y₁) and has a coefficient of direction k reads:
y – y₁ = k (x – x₁)
The equation of a line passing through two given points MZ (x₁, y₁) and M₂ (x₂, y₂) has the following form:
6. Distance from point to line
The distance from the point M (x₀, y₀) to the line Ax + By + C = 0 is calculated by the formula:
7. Angle between two lines
The angle between the lines l₁, and l₂ (Fig. B) is determined by the formula:
where k₁, and k₂ are coefficients of the directions of the lines l₁ and l₂, respectively.
From formula (1) are obtained the conditions for parallelism and normality of two lines. Thus, l₁ || l₂, if k₂ = k₁, l₁⊥l₂ if 1 + k₁k₂ = 0