The general equation of a straight line and its analysis.

у

М₂(0;b)

b

M₁(a;0)

0 a x

Knowing the two points М ₁ and М ₂ through which the line passes, we can write the equation of the line in form (10):

, or .

This is the two-intercept equation of the line.

Example. Reduce the equation , to the two-intercept form.

Take the variables to the left-hand side

.

We have

, where and

The angle between two straight lines. Parallel and perpendicular lines. Suppose given two straight lines with slopes and .

у j=?

j ₁ j ₂

j ₁ j ₂

0 х

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:

. (11)

Formula (11) gives conditions for two lines to be parallel and perpendicular.

(1) Suppose that the right lines are parallel, i.e., the angle between them is ; substituting it into formula (11), 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.

(2) Suppose that two straight lines are perpendicular; then the angle between them is . Substituting it in (11), we obtain

.

This fraction equals infinity when the denominator vanishes:

.

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

.

Example. Write equations of the straight lines passing through the point М₀ (1;1) and parallel and perpendicular to the line .

Let us write the equation of the given line in the form y=kx+b:

, or ; .

To compose an equation of a straight line, we use the formula

. (*)

Since the required line must be parallel to the required one, it follows that

.

Substituting this into equation (*), we obtain

, or .

The perpendicularity condition gives the slope:

.

The equation of the required line 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:

; ;

. (12)

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:

; ;

;

; ; ; ;

. (12΄)

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:

; ;

. (12΄΄)

This is the condition for straight lines to coincide.

Example. 1. Concider the straight lines given by

According to formula (12), we have , which means that these lines intersect.

2. Consider the straight lines given by

According to formula (12΄), we have , and the lines are parallel.

3. Consider the straight lines given by

According to formula (12΄΄) we have , and the lines coincide.