The general equation of a straight line and its analysis

Definition 2. A first–order equation in variables x and y determines a straight line in the plane.

The general equation has the form ,

where А and В are called the coefficients of the variables.

1. If the free term is С=0, then the equation has the form .

Since х=0 and y=0 satisfy this equation, it follows that the straight line passes through the origin.

2. If the coefficient of х is А=0, then the equation has the form

or , i.e., the line is parallel to the x –axis.

3. If the coefficient of y is B=0, then the equation has the form

,

and the line is parallel to the y– axis.

4. If А=С=0, then the line В у=0, (or у=0) coincides with the x– axis.

5. If В=С=0, then the line А х=0, (or х=0) coincides with the y– axis.

The angle between two straight lines. Parallel and perpendicular lines. Suppose given two straight lines with slopes and . It is seen from the figure that the angle between the lines equals j=j₂–j₁. Using the formula for the tangent of the difference between two angels, we obtain .

Replacing the tangents by the slopes k₁ and k_{2 ,} we obtain the following formula for the tangent of the angle between two straight lines:

. (1)

(*) Suppose that the right lines are parallel, i.e., the angle between them is ; substituting it into formula (1), we obtain

.

This fraction vanishes, if k₂–k₁=0.

Thus, two straight lines are parallel if and only if their slopes are equal:

k ₂= k ₁.

(**) Suppose that two straight lines are perpendicular; then the angle between them is . Substituting it in (1), we obtain

.

This fraction equals infinity when the denominator vanishes:

.

Consequently, the condition for two straight lines to be perpendicular is

.

The mutual arrangement of two straight lines. Given equations of two straight lines

and .

Determine conditions on the coefficient, for these right lines to intersect, be parallel, or coincide.

1. To determine the mutual arrangement of lines, we must analyze the system of equations

If the lines intersect, then this system has a unique solution, and its principal determinant is nonzero:

; ;

.

Thus, if the straight lines intersect, then the coefficients must not be proportional.

2. Suppose that the straight lines are parallel, i.e., they have no common points, and the system of equations has no solution; then the principal determinant vanishes, and the auxiliary determinants are nonzero:

; ;

;

; ; ; ;

.

This is the parallelis condition.

3. When the straight lines coincide, i.e., have many common points, the system of equations has infinitely many solutions. In this case, the auxiliary and principal determinants are zero:

; ;

.

This is the condition for straight lines to coincide.

Distance d from the point M₀(x₀,y₀) to the line is the length of the perpendicular from this point on the line.

The distance d is defined by:

.

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