Equation of a straight line

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.

Properties:

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).

315-c4
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:

316-f1

in which a and b are segments that intersect the lines on the coordinate axes x and y, respectively (fig. 5).

317-c5


Fig.5

4. Normal type equation of a line

The normal type of equation of a line given in general is

317-f1

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:

318-f2

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:

319-f2

7. Angle between two lines

The angle between the lines l₁, and l₂ (Fig. B) is determined by the formula:

320-f1

where k₁, and k₂ are coefficients of the directions of the lines l₁ and l₂, respectively.

320-c6
Fig. 6

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

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