Equation of a line passing through 2. Equation of a line - types of equation of a line: passing through a point, general, canonical, parametric, etc.

Definition. In the Cartesian rectangular coordinate system, a vector with components (A, B) is perpendicular to the straight line given by the equation Ax + By + C = 0.

Example. Find the equation of the line passing through the point A(1, 2) perpendicular to the vector (3, -1).

Solution. With A = 3 and B = -1, let’s compose the equation of the straight line: 3x – y + C = 0. To find the coefficient C, we substitute the coordinates of the given point A into the resulting expression. We get: 3 – 2 + C = 0, therefore C = -1. Total: the required equation: 3x – y – 1 = 0.

Equation of a line passing through two points

Let two points M 1 (x 1, y 1, z 1) and M 2 (x 2, y 2, z 2) be given in space, then the equation of the line passing through these points is:

If any of the denominators is equal to zero, the corresponding numerator should be equal to zero. On the plane, the equation of the line written above is simplified:

if x 1 ≠ x 2 and x = x 1, if x 1 = x 2.

The fraction = k is called slope straight.

Example. Find the equation of the line passing through points A(1, 2) and B(3, 4).

Solution. Applying the formula written above, we get:

Equation of a straight line from a point and slope

If the general equation of the straight line Ax + By + C = 0 is reduced to the form:

and designate , then the resulting equation is called equation of a straight line with slopek.

Equation of a straight line from a point and a direction vector

By analogy with the point considering the equation of a straight line through a normal vector, you can enter the definition of a straight line through a point and the directing vector of the straight line.

Definition. Each non-zero vector (α 1, α 2), the components of which satisfy the condition A α 1 + B α 2 = 0 is called a directing vector of the line

Ax + Wu + C = 0.

Example. Find the equation of a straight line with a direction vector (1, -1) and passing through the point A(1, 2).

Solution. We will look for the equation of the desired line in the form: Ax + By + C = 0. In accordance with the definition, the coefficients must satisfy the conditions:

1 * A + (-1) * B = 0, i.e. A = B.

Then the equation of the straight line has the form: Ax + Ay + C = 0, or x + y + C / A = 0. for x = 1, y = 2 we obtain C/ A = -3, i.e. required equation:

Equation of a line in segments

If in the general equation of the straight line Ах + Ву + С = 0 С≠0, then, dividing by –С, we get: or

The geometric meaning of the coefficients is that the coefficient A is the coordinate of the point of intersection of the line with the Ox axis, and b – the coordinate of the point of intersection of the straight line with the Oy axis.

Example. The general equation of the line x – y + 1 = 0 is given. Find the equation of this line in segments.

C = 1, , a = -1, b = 1.

Normal equation of a line

If both sides of the equation Ax + By + C = 0 are divided by the number which is called normalizing factor, then we get

xcosφ + ysinφ - p = 0 –

normal equation of a line. The sign ± of the normalizing factor must be chosen so that μ * C< 0. р – длина перпендикуляра, опущенного из начала координат на прямую, а φ - угол, образованный этим перпендикуляром с положительным направлением оси Ох.

Example. The general equation of the line 12x – 5y – 65 = 0 is given. It is required to write various types of equations for this line.

equation of this line in segments:

equation of this line with slope: (divide by 5)

normal equation of a line:

; cos φ = 12/13; sin φ= -5/13; p = 5.

C It should be noted that not every line can be represented by an equation in segments, for example, lines parallel to axes or passing through the origin.

Example. The straight line cuts off equal positive segments on the coordinate axes. Write an equation for a straight line if the area of ​​the triangle formed by these segments is 8 cm 2.

Solution. The equation of the straight line has the form: , ab /2 = 8; a = 4; -4. a = -4 is not suitable according to the conditions of the problem. Total: or x + y – 4 = 0.

Example. Write an equation for a straight line passing through point A(-2, -3) and the origin.

Solution. The equation of the straight line is: , where x 1 = y 1 = 0; x 2 = -2; y2 = -3.

Let's look at how to create an equation for a line passing through two points using examples.

Example 1.

Write an equation for a straight line passing through points A(-3; 9) and B(2;-1).

Method 1 - create an equation of a straight line with an angle coefficient.

The equation of a straight line with an angular coefficient has the form . Substituting the coordinates of points A and B into the equation of the straight line (x= -3 and y=9 - in the first case, x=2 and y= -1 - in the second), we obtain a system of equations from which we find the values ​​of k and b:

Adding the 1st and 2nd equations term by term, we get: -10=5k, whence k= -2. Substituting k= -2 into the second equation, we find b: -1=2·(-2)+b, b=3.

Thus, y= -2x+3 is the required equation.

Method 2 - let's create a general equation of a straight line.

The general equation of a straight line has the form . Substituting the coordinates of points A and B into the equation, we obtain the system:

Since the number of unknowns is greater than the number of equations, the system is not solvable. But all variables can be expressed through one. For example, through b.

By multiplying the first equation of the system by -1 and adding term by term with the second:

we get: 5a-10b=0. Hence a=2b.

Let's substitute the resulting expression into the second equation: 2·2b -b+c=0; 3b+c=0; c= -3b.
Substitute a=2b, c= -3b into the equation ax+by+c=0:

2bx+by-3b=0. It remains to divide both sides by b:

The general equation of a straight line can easily be reduced to the equation of a straight line with an angular coefficient:

Method 3 - create an equation of a straight line passing through 2 points.

The equation of a line passing through two points is:

Let's substitute the coordinates of points A(-3; 9) and B(2;-1) into this equation

(that is, x 1 = -3, y 1 =9, x 2 =2, y 2 = -1):

and simplify:

whence 2x+y-3=0.

In school courses, the equation of a straight line with an angle coefficient is most often used. But the easiest way is to derive and use the formula for the equation of a line passing through two points.

Comment.

If, when substituting the coordinates of given points, one of the denominators of the equation

turns out to be equal to zero, then the required equation is obtained by equating the corresponding numerator to zero.

Example 2.

Write an equation for a straight line passing through two points C(5; -2) and D(7;-2).

We substitute the coordinates of points C and D into the equation of a straight line passing through 2 points.

This article is part of the topic equation of a line in a plane. Here we will look at it from all sides: we will start with the proof of the theorem that specifies the form of the general equation of a line, then we will consider an incomplete general equation of a line, we will give examples of incomplete equations of a line with graphic illustrations, and in conclusion we will dwell on the transition from a general equation of a line to other types of equations of this line and give detailed solutions to typical problems for composing the general equation of a straight line.

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General equation of a straight line - basic information.

Let's analyze this algorithm when solving an example.

Example.

Write parametric equations of a line that is given by the general equation of a line .

Solution.

First, we reduce the original general equation of the line to the canonical equation of the line:

Now we take the left and right sides of the resulting equation to be equal to the parameter. We have

Answer:

From a general equation of a straight line, it is possible to obtain an equation of a straight line with an angle coefficient only when . What do you need to do to make the transition? Firstly, on the left side of the general straight line equation, leave only the term , the remaining terms must be moved to the right side with the opposite sign: . Secondly, divide both sides of the resulting equality by the number B, which is non-zero, . That's all.

Example.

A straight line in a rectangular coordinate system Oxy is given by the general equation of a straight line. Get the equation of this line with the slope.

Solution.

Let's carry out the necessary actions: .

Answer:

When a line is given by the complete general equation of the line, it is easy to obtain the equation of the line in segments of the form. To do this, we transfer the number C to the right side of the equality with the opposite sign, divide both sides of the resulting equality by –C, and finally transfer the coefficients for the variables x and y to the denominators:

Consider the equation of a straight line passing through a point and a normal vector. Let a point and a nonzero vector be specified in the coordinate system (Fig. 1).

Definition

As we can see, there is a single straight line that passes through the point perpendicular to the direction of the vector (in this case it is called normal vector straight ).

Rice. 1

Let us prove that the linear equation

this is an equation of a line, that is, the coordinates of each point of the line satisfy equation (1), but the coordinates of a point that does not lie on do not satisfy equation (1).

To prove this, let us note that the scalar product of vectors and = in coordinate form coincides with the left side of equation (1).

Next we use the obvious property of the line: the vectors and are perpendicular if and only if the point lies on . And provided that both vectors are perpendicular, their scalar product (2) turns into for all points that lie on, and only for them. This means (1) is the equation of the straight line.

Definition

Equation (1) is called equation of the line that passes through a given pointwith normal vector = .

Let's transform equation (1)

Denoting = , we get

Thus, a linear equation of the form (3) corresponds to a straight line. On the contrary, using a given equation of the form (3), where at least one of the coefficients is not equal to zero, a straight line can be constructed.

Indeed, let a pair of numbers satisfy equation (3), that is

Subtracting the latter from (3), we obtain the relation that determines the straight line behind the vector and the point.

Study of the general equation of the line

It is useful to know the features of placing a line in certain cases when one or two of the numbers are equal to zero.

1. The general equation looks like this: . The point satisfies it, which means the line passes through the origin. It can be written: = – x (see Fig. 2).

Rice. 2

We believe that:

If we put , then , we get another point (see Fig. 2).

2. , then the equation looks like this, where = –. The normal vector lies on the axis, a straight line. Thus, the straight line is perpendicular at point , or parallel to the axis (see Fig. 3). In particular, if and , then and the equation is the equation of the ordinate axis.

Rice. 3

3. Similarly, when the equation is written, where . The vector belongs to the axis. Straight line at a point (Fig. 4).

If, then the equation of the axis is .

The study can be formulated in this form: the straight line is parallel to the coordinate axis, the change of which is absent in the general equation of the straight line.

For example:

Let's construct a straight line using the general equation, provided that - are not equal to zero. To do this, it is enough to find two points that lie on this line. It is sometimes more convenient to find such points on coordinate axes.

Let us then = –.

When , then = –.

Let us denote – = , – = . Points and were found. Let us plot and draw a straight line on the axes and through them (see Fig. 5).

Rice. 5

From the general, you can move on to an equation that will include the numbers and:

And then it turns out:

Or, according to the notation, we obtain the equation

Which is called equation of a straight line in segments. The numbers and, accurate to the sign, are equal to the segments that are cut off by a straight line on the coordinate axes.

Equation of a straight line with slope

To find out what the equation of a straight line with a slope is, consider equation (1):

Denoting – = , we get

equation of a line that passes through a point in a given direction. The geometric content of the coefficient is clear from Fig. 6.

B = = , where is the smallest angle by which the positive direction of the axis needs to be rotated around the common point until it aligns with the straight line. Obviously, if the angle is acute, then title="Rendered by QuickLaTeX.com" height="17" width="97" style="vertical-align: -4px;">; если же – тупой угол, тогда .!}

Let's open the brackets in (5) and simplify it:

Where . Relationship (6) – equation straight line with slope. When , is a segment that cuts off a straight line on the axis (see Fig. 6).

Note!

To move from a general straight line equation to an equation with a slope coefficient, you must first solve for .

Rice. 6

= – x + – =

where denoted = –, = –. If, then from the study of the general equation it is already known that such a straight line is perpendicular to the axis.

Let's look at the canonical equation of a straight line using an example.

Let a point and a nonzero vector be specified in the coordinate system (Fig. 7).

Rice. 7

It is necessary to create an equation for a straight line that passes through a point parallel to the vector, which is called the direction vector. An arbitrary point belongs to this line if and only if . Since the vector is given, and the vector is , then, according to the parallelism condition, the coordinates of these vectors are proportional, that is:

Definition

Relationship (7) is called the equation of a line that passes through a given point in a given direction or the canonical equation of a line.

Let us note that we can move to an equation of the form (7), for example, from the equation of a pencil of lines (4)

or from the equation of a straight line through a point and a normal vector (1):

It was assumed above that the direction vector is non-zero, but it may happen that one of its coordinates, for example, . Then expression (7) will be formally written:

which doesn't make sense at all. However, we accept and obtain the equation of the straight line perpendicular to the axis. Indeed, from the equation it is clear that the straight line is defined by a point and a direction vector perpendicular to the axis. If we remove the denominator from this equation, then we get:

Or - the equation of a straight line perpendicular to the axis. A similar result would be obtained for the vector .

Parametric equation of a line

To understand what a parametric equation of a line is, you need to return to equation (7) and equate each fraction (7) to a parameter. Since at least one of the denominators in (7) is not equal to zero, and the corresponding numerator can acquire arbitrary values, then the region of parameter change is the entire numerical axis.

Definition

Equation (8) is called the parametric equation of a straight line.

Examples of straight line problems

Of course, it is difficult to solve anything solely based on definitions, because you need to solve at least a few examples or problems on your own that will help consolidate the material you have covered. Therefore, let's analyze the main tasks in a straight line, since similar problems often come across in exams and tests.

Canonical and parametric equation

Example 1

On a straight line given by the equation, find a point that is located at a distance of 10 units from the point of this straight line.

Solution:

Let sought after point of a straight line, then for the distance we write . Given that . Since the point belongs to a line that has a normal vector, then the equation of the line can be written: = = and then it turns out:

Then the distance . Subject to , or . From the parametric equation:

Example 2

Task

The point moves uniformly with speed in the direction of the vector from the starting point. Find the coordinates of the point through from the beginning of the movement.

Solution

First you need to find the unit vector. Its coordinates are direction cosines:

Then the velocity vector:

X = x = .

The canonical equation of the line will now be written:

= = , = – parametric equation. After this, you need to use the parametric equation of the straight line at .

Solution:

The equation of a line that passes through a point is found using the formula for a pencil of lines, where slope for a straight line and = for a straight line.

Considering the figure, where you can see that between straight lines and - there are two angles: one is acute, and the second is obtuse. According to formula (9), this is the angle between the straight lines and by which you need to rotate the straight line counterclockwise relative to their intersection point until it aligns with the straight line .

So, we remembered the formula, we figured out the angles and now we can return to our example. This means, taking into account formula (9), we first find the equations of the leg.

Since rotating the straight line by an angle counterclockwise relative to the point leads to alignment with the straight line, then in formula (9) , a . From the equation:

Using the beam formula, the equation of a straight line will be written:

Similarly we find , and ,

Line equation:

Equation of a line – types of equation of a line: passing through a point, general, canonical, parametric, etc. updated: November 22, 2019 by: Scientific Articles.Ru

Properties of a straight line in Euclidean geometry.

An infinite number of straight lines can be drawn through any point.

Through any two non-coinciding points a single straight line can be drawn.

Two divergent lines in a plane either intersect at a single point or are

parallel (follows from the previous one).

In three-dimensional space, there are three options for the relative position of two lines:

• lines intersect;

• lines are parallel;

• straight lines intersect.

Straight line— algebraic curve of the first order: a straight line in the Cartesian coordinate system

is given on the plane by an equation of the first degree (linear equation).

General equation of a straight line.

Definition. Any straight line on the plane can be specified by a first-order equation

Ax + Wu + C = 0,

and constant A, B are not equal to zero at the same time. This first order equation is called general

equation of a straight line. Depending on the values ​​of the constants A, B And WITH The following special cases are possible:

. C = 0, A ≠0, B ≠ 0- a straight line passes through the origin

. A = 0, B ≠0, C ≠0 (By + C = 0)- straight line parallel to the axis Oh

. B = 0, A ≠0, C ≠ 0 (Ax + C = 0)- straight line parallel to the axis OU

. B = C = 0, A ≠0- the straight line coincides with the axis OU

. A = C = 0, B ≠0- the straight line coincides with the axis Oh

The equation of a straight line can be presented in different forms depending on any given

initial conditions.

Equation of a straight line from a point and normal vector.

Definition. In a Cartesian rectangular coordinate system, a vector with components (A, B)

perpendicular to the line given by the equation

Ax + Wu + C = 0.

Example. Find the equation of a line passing through a point A(1, 2) perpendicular to the vector (3, -1).

Solution. With A = 3 and B = -1, let’s compose the equation of the straight line: 3x - y + C = 0. To find the coefficient C

Let's substitute the coordinates of the given point A into the resulting expression. We get: 3 - 2 + C = 0, therefore

C = -1. Total: the required equation: 3x - y - 1 = 0.

Equation of a line passing through two points.

Let two points be given in space M 1 (x 1 , y 1 , z 1) And M2 (x 2, y 2, z 2), Then equation of a line,

passing through these points:

If any of the denominators is zero, the corresponding numerator should be set equal to zero. On

plane, the equation of the straight line written above is simplified:

If x 1 ≠ x 2 And x = x 1, If x 1 = x 2 .

Fraction = k called slope straight.

Example. Find the equation of the line passing through points A(1, 2) and B(3, 4).

Solution. Applying the formula written above, we get:

Equation of a straight line using a point and slope.

If the general equation of the line Ax + Wu + C = 0 lead to:

and designate , then the resulting equation is called

equation of a straight line with slope k.

Equation of a straight line from a point and a direction vector.

By analogy with the point considering the equation of a straight line through the normal vector, you can enter the task

a straight line through a point and a directing vector of a straight line.

Definition. Every non-zero vector (α 1 , α 2), whose components satisfy the condition

Aα 1 + Bα 2 = 0 called directing vector of a straight line.

Ax + Wu + C = 0.

Example. Find the equation of a straight line with a direction vector (1, -1) and passing through the point A(1, 2).

Solution. We will look for the equation of the desired line in the form: Ax + By + C = 0. According to the definition,

coefficients must satisfy the following conditions:

1 * A + (-1) * B = 0, i.e. A = B.

Then the equation of the straight line has the form: Ax + Ay + C = 0, or x + y + C / A = 0.

at x = 1, y = 2 we get C/A = -3, i.e. required equation:

x + y - 3 = 0

Equation of a straight line in segments.

If in the general equation of the straight line Ах + Ву + С = 0 С≠0, then, dividing by -С, we get:

or where

The geometric meaning of the coefficients is that the coefficient a is the coordinate of the intersection point

straight with axis Oh, A b- coordinate of the point of intersection of the line with the axis OU.

Example. The general equation of a straight line is given x - y + 1 = 0. Find the equation of this line in segments.

C = 1, , a = -1, b = 1.

Normal equation of a line.

If both sides of the equation Ax + Wu + C = 0 divide by number which is called

normalizing factor, then we get

xcosφ + ysinφ - p = 0 -normal equation of a line.

The sign ± of the normalizing factor must be chosen so that μ*C< 0.

R- the length of the perpendicular dropped from the origin to the straight line,

A φ - the angle formed by this perpendicular with the positive direction of the axis Oh.

Example. The general equation of the line is given 12x - 5y - 65 = 0. Required to write different types of equations

this straight line.

The equation of this line in segments:

The equation of this line with the slope: (divide by 5)

Equation of a line:

cos φ = 12/13; sin φ= -5/13; p = 5.

It should be noted that not every straight line can be represented by an equation in segments, for example, straight lines,

parallel to the axes or passing through the origin.

The angle between straight lines on a plane.

Definition. If two lines are given y = k 1 x + b 1 , y = k 2 x + b 2, then the acute angle between these lines

will be defined as

Two lines are parallel if k 1 = k 2. Two lines are perpendicular

If k 1 = -1/ k 2 .

Theorem.

Direct Ax + Wu + C = 0 And A 1 x + B 1 y + C 1 = 0 parallel when the coefficients are proportional

A 1 = λA, B 1 = λB. If also С 1 = λС, then the lines coincide. Coordinates of the point of intersection of two lines

are found as a solution to the system of equations of these lines.

The equation of a line passing through a given point perpendicular to a given line.

Definition. Line passing through a point M 1 (x 1, y 1) and perpendicular to the line y = kx + b

represented by the equation:

Distance from a point to a line.

Theorem. If a point is given M(x 0, y 0), then the distance to the straight line Ax + Wu + C = 0 defined as:

Proof. Let the point M 1 (x 1, y 1)- the base of a perpendicular dropped from a point M for a given

direct. Then the distance between points M And M 1:

(1)

Coordinates x 1 And at 1 can be found as a solution to the system of equations:

The second equation of the system is the equation of a straight line passing through a given point M 0 perpendicularly

given straight line. If we transform the first equation of the system to the form:

A(x - x 0) + B(y - y 0) + Ax 0 + By 0 + C = 0,

then, solving, we get:

Substituting these expressions into equation (1), we find:

The theorem has been proven.