Determine which line on the plane is given by the equation. Equation of a straight line, types of equation of a straight line on a plane
Consider the function given by the formula (equation)
This function, and hence equation (11), corresponds on the plane to a well-defined line, which is the graph of this function (see Fig. 20). It follows from the definition of the function graph that this line consists of those and only those points of the plane whose coordinates satisfy equation (11).
Let now
The line, which is the graph of this function, consists of those and only those points of the plane whose coordinates satisfy equation (12). This means that if a point lies on the specified line, then its coordinates satisfy equation (12). If the point does not lie on this line, then its coordinates do not satisfy equation (12).
Equation (12) is resolved with respect to y. Consider an equation containing x and y that is not resolved with respect to y, such as the equation
Let us show that a line corresponds to this equation in the plane, namely, a circle centered at the origin of coordinates and with a radius equal to 2. Let us rewrite the equation in the form
Its left side is the square of the point's distance from the origin (see § 2, item 2, formula 3). From equality (14) it follows that the square of this distance is 4.
This means that any point whose coordinates satisfy equation (14), and hence equation (13), is located at a distance of 2 from the origin.
The locus of such points is a circle centered at the origin and radius 2. This circle will be the line corresponding to equation (13). The coordinates of any of its points obviously satisfy equation (13). If the point does not lie on the circle we found, then the square of its distance from the origin will either be greater or less than 4, which means that the coordinates of such a point do not satisfy equation (13).
Let now, in the general case, given the equation
on the left side of which is an expression containing x and y.
Definition. The line defined by equation (15) is the locus of points in the plane whose coordinates satisfy this equation.
This means that if the line L is determined by the equation, then the coordinates of any point of L satisfy this equation, and the coordinates of any point of the plane lying outside L do not satisfy equation (15).
Equation (15) is called the line equation
Comment. It should not be thought that any equation defines any line. For example, the equation does not define any line. Indeed, for any real values of and y, the left side of this equation is positive, and the right side is equal to zero, and therefore, this equation cannot satisfy the coordinates of any point in the plane
A line can be defined on a plane not only by an equation containing Cartesian coordinates, but also by an equation in polar coordinates. The line defined by the equation in polar coordinates is the locus of points in the plane whose polar coordinates satisfy this equation.
Example 1. Construct the Archimedes spiral at .
Solution. Let's make a table for some values of the polar angle and the corresponding values of the polar radius.
We build a point in the polar coordinate system, which, obviously, coincides with the pole; then, drawing the axis at an angle to the polar axis, we construct a point with a positive coordinate on this axis; after that, we similarly construct points with positive values of the polar angle and polar radius (the axes for these points are not indicated in Fig. 30).
As is known, any point on the plane is determined by two coordinates in some coordinate system. Coordinate systems can be different depending on the choice of basis and origin.
Definition: The equation of a line is the relation y = f(x) between the coordinates of the points that make up this line.
Note that the line equation can be expressed in a parametric way, that is, each coordinate of each point is expressed through some independent parameter t. A typical example is the trajectory of a moving point. In this case, time plays the role of a parameter.
Different types of equation of a straight line
General equation of a straight line.
Any line in the plane can be given by a first order equation
Ah + Wu + C = 0,
moreover, the constants A, B are not equal to zero at the same time, i.e. A 2 + B 2 ¹ 0. This first-order equation is called the general equation of the straight line .
Depending on the values constant A, B and C, the following special cases are possible:
C \u003d 0, A ¹ 0, B ¹ 0 - the line passes through the origin
A \u003d 0, B ¹ 0, C ¹ 0 ( By + C \u003d 0) - the line is parallel to the Ox axis
B \u003d 0, A ¹ 0, C ¹ 0 (Ax + C \u003d 0) - the line is parallel to the Oy axis
B \u003d C \u003d 0, A ¹ 0 - the straight line coincides with the Oy axis
A \u003d C \u003d 0, B ¹ 0 - the straight line coincides with the Ox axis
The equation of a straight line can be represented in various forms depending on any given initial conditions.
Equation of a straight 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 a straight line passing through these points:
If any of the denominators is equal to zero, the corresponding numerator should be set equal to zero. On a plane, the equation of a straight line written above is simplified:
if x 1 ¹ x 2 and x \u003d x 1, if x 1 \u003d x 2.
The fraction = k is called the slope of the straight line.
Equation of a straight line by a point and a slope.
If the general equation of the straight line Ax + Vy + C = 0 lead to the form:
and denote , then the resulting equation is called the equation of a straight line with slope k.
Equation of a straight line in segments.
If in the general equation of the straight line Ah + Vu + С = 0 С ¹ 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 x-axis, and b- the coordinate of the point of intersection of the straight line with the Oy axis.
Normal equation of a straight line.
If both parts of the equation Ax + Vy + C = 0 are divided by the number , which is called the normalizing factor, then we get
xcosj + ysinj - p = 0 –
normal equation of a straight line.
The sign ± of the normalizing factor must be chosen so that m × С< 0.
p is the length of the perpendicular dropped from the origin to the straight line, and j is the angle formed by this perpendicular with the positive direction of the Ox axis.
Angle between lines on a plane.
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. The straight lines Ax + Vy + C \u003d 0 and A 1 x + B 1 y + C 1 \u003d 0 are parallel when the coefficients A 1 \u003d lA, B 1 \u003d lB are proportional. If also C 1 = lC, then the lines coincide.
The coordinates of the point of intersection of two lines are found as a solution to a system of two equations.
The distance from a point to a line.
Theorem. If a point M(x 0, y 0) is given, then the distance to the line Ax + Vy + C \u003d 0 is defined as
Lecture 5
Introduction to analysis. Differential calculus of a function of one variable.
FUNCTION LIMIT
Limit of a function at a point.
0 a - D a a + D x
Figure 1. Limit of a function at a point.
Let the function f(x) be defined in some neighborhood of the point x = a (that is, at the point x = a itself, the function may not be defined)
Definition. The number A is called the limit of the function f(x) for x®a if for any e>0 there exists a number D>0 such that for all x such that
0 < ïx - aï < D
the inequality ïf(x) - Aï< e.
The same definition can be written in a different form:
If a - D< x < a + D, x ¹ a, то верно неравенство А - e < f(x) < A + e.
Writing the limit of a function at a point:
Definition.
If f(x) ® A 1 for x ® a only for x< a, то - называется пределом функции f(x) в точке х = а слева, а если f(x) ® A 2 при х ® а только при x >a, then it is called the limit of the function f(x) at the point x = a on the right.
The above definition refers to the case when the function f(x) is not defined at the point x = a itself, but is defined in some arbitrarily small neighborhood of this point.
Limits A 1 and A 2 are also called unilateral outside the function f(x) at the point x = a. It is also said that A function limit f(x).
Equation of a line on a plane.
As is known, any point on the plane is determined by two coordinates in some coordinate system. Coordinate systems can be different depending on the choice of basis and origin.
Definition. Line equation is called the ratio y=f(x ) between the coordinates of the points that make up this line.
Note that the line equation can be expressed in a parametric way, that is, each coordinate of each point is expressed through some independent parametert.
A typical example is the trajectory of a moving point. In this case, time plays the role of a parameter.
Equation of a straight line on a plane.
Definition. Any line in the plane can be given by a first order equation
Ah + Wu + C = 0,
moreover, the constants A, B are not equal to zero at the same time, i.e. A 2 + B 2¹ 0. This first order equation is called the general equation of a straight line.
Depending on the values of the constants A, B and C, the following special cases are possible:
C = 0, A ¹ 0, B ¹ 0 - the line passes through the origin
A = 0, B ¹ 0, C ¹ 0 ( By + C \u003d 0) - a straight line is parallel to the Ox axis
B = 0, A ¹ 0, C ¹ 0 ( Ax + C = 0) - a straight line parallel to the Oy axis
B \u003d C \u003d 0, A ¹ 0 - the line coincides with the Oy axis
A = C = 0, B ¹ 0 - the line coincides with the Ox axis
The equation of a straight line can be presented in various forms depending on any given initial conditions.
The distance from a point to a line.
Theorem. If a point M(x 0, y 0) is given, then the distance to the line Ax + Vy + C \u003d 0 is defined as
.
Proof. Let the point M 1 (x 1, y 1) be the base of the perpendicular dropped from the point M to the given line. Then the distance between points M and M 1:
(1)
Coordinates x 1 and y 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 perpendicular to a 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.
Example. Determine the angle between the lines: y=-3x+7; y = 2 x + 1.
K 1 \u003d -3; k 2 = 2tg j = ; j = p /4.
Example. Show that the lines 3x - 5y + 7 = 0 and 10x + 6y - 3 = 0 are perpendicular.
Find: k 1 = 3/5, k 2 = -5/3, k 1 k 2 = -1, hence the lines are perpendicular.
Example. Given the vertices of the triangle A(0; 1), B(6;5), C (12; -1). Find the equation for the height drawn from vertex C.
In the last article, we considered the main points regarding the topic of a straight line on a plane. Now let's move on to studying the equation of a straight line: consider which equation can be called an equation of a straight line, and also what form the equation of a straight line has in a plane.
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Definition of the equation of a straight line in a plane
Let's say that there is a straight line, which is given in a rectangular Cartesian coordinate system O x y.
Definition 1
Straight line- This geometric figure, which is made up of dots. Each point has its own coordinates along the abscissa and ordinate axes. The equation that describes the dependence of the coordinates of each point of a straight line in the Cartesian system O x y is called the equation of a straight line on a plane.
In fact, the equation of a straight line in a plane is an equation with two variables, which are denoted as x and y. The equation turns into an identity when the values of any of the points of the straight line are substituted into it.
Let's see what form the equation of a straight line in a plane will have. This will be the focus of the next section of our article. Note that there are several options for writing the equation of a straight line. This is explained by the presence of several ways to set a straight line on a plane, and also by the different specifics of the tasks.
Let's get acquainted with the theorem that defines the form of the equation of a straight line on a plane in the Cartesian coordinate system O x y .
Theorem 1
An equation of the form A x + B y + C = 0 , where x and y are variables, and A, B and C are some real numbers, of which A and B are not equal to zero, defines a straight line in the Cartesian coordinate system O x y . In turn, any straight line on the plane can be given by an equation of the form A x + B y + C = 0 .
Thus, the general equation of a straight line in the plane has the form A x + B y + C = 0 .
Let us explain some important aspects of the topic.
Example 1
Look at the drawing.
The line in the drawing is determined by an equation of the form 2 x + 3 y - 2 \u003d 0, since the coordinates of any point that makes up this line satisfy the above equation. At the same time, a certain number of points in the plane, defined by the equation 2 x + 3 y - 2 = 0, give us the straight line that we see in the figure.
The general equation of a straight line can be complete or incomplete. In the complete equation, all numbers A, B and C are non-zero. In all other cases, the equation is considered incomplete. An equation of the form A x + B y = 0 defines a straight line that passes through the origin. If A is zero, then the equation A x + B y + C = 0 defines a straight line parallel to the x-axis O x . If B is equal to zero, then the line is parallel to the ordinate axis O y .
Conclusion: for a certain set of values of numbers A, B and C, using the general equation of a straight line, you can write any straight line on a plane in a rectangular coordinate system O x y.
The line given by an equation of the form A x + B y + C = 0 has a normal line vector with coordinates A , B .
All the given equations of lines, which we will consider below, can be obtained from the general equation of a line. The reverse process is also possible, when any of the considered equations can be reduced to the general equation of a straight line.
You can understand all the nuances of the topic in the article "The general equation of a straight line." In the material we provide a proof of the theorem with graphic illustrations and a detailed analysis of examples. Particular attention is paid to the transitions from the general equation of a straight line to equations of other types and vice versa.
The equation of a straight line in segments has the form x a + y b = 1 , where a and b are some real numbers that are not equal to zero. The absolute values of the numbers a and b are equal to the length of the segments that are cut off by a straight line on the coordinate axes. The length of the segments is measured from the origin of coordinates.
Thanks to the equation, you can easily draw a straight line on the drawing. To do this, it is necessary to mark points a, 0 and 0, b in a rectangular coordinate system, and then connect them with a straight line.
Example 2
Let's build a straight line, which is given by the formula x 3 + y - 5 2 = 1. We mark two points on the graph 3 , 0 , 0 , - 5 2 , connect them together.
These equations, having the form y = k · x + b, should be well known to us from the course of algebra. Here x and y are variables, k and b are some real numbers, of which k is the slope. In these equations, the variable y is a function of the argument x.
Let's give the definition of the slope through the definition of the angle of inclination of the straight line to the positive direction of the axis O x .
Definition 2
To denote the angle of inclination of the straight line to the positive direction of the axis O x in the Cartesian coordinate system, we introduce the value of the angle α. The angle is measured from the positive direction of the x-axis to a straight line counterclockwise. The angle α is considered equal to zero if the line is parallel to the O x axis or coincides with it.
The slope of a straight line is the tangent of the slope of that straight line. It is written as follows k = t g α . For a straight line that is parallel to the axis O y or coincides with it, it is not possible to write the equation of a straight line with a slope, since the slope in this case turns into infinity (does not exist).
The straight line, which is given by the equation y = k x + b, passes through the point 0, b on the y-axis. This means that the equation of a straight line with a slope y \u003d k x + b sets a straight line on the plane that passes through the point 0, b and forms an angle α with the positive direction of the O x axis, and k \u003d t g α.
Example 3
Let's draw a straight line, which is defined by an equation of the form y = 3 · x - 1 .
This line must pass through the point (0 , - 1) . The angle of inclination α = a r c t g 3 = π 3 is equal to 60 degrees to the positive direction of the O x axis. Slope is 3
Please note that using the equation of a straight line with a slope it is very convenient to look for the equation of a tangent to the graph of a function at a point.
More material on the topic can be found in the article "The Equation of a Line with a Slope". In addition to theory, there are a large number of graphic examples and a detailed analysis of tasks.
This type of equation has the form x - x 1 a x \u003d y - y 1 a y, where x 1, y 1, a x, a y are some real numbers, of which a x and a y are not equal to zero.
The straight line given by the canonical equation of the straight line passes through the point M 1 (x 1 , y 1) . The numbers a x and a y in the denominators of the fractions are the coordinates of the direction vector of the straight line. This means that the canonical equation of a straight line x - x 1 a x = y - y 1 a y in the Cartesian coordinate system O x y corresponds to a line passing through the point M 1 (x 1 , y 1) and having a direction vector a → = (a x , a y) .
Example 4
Draw a straight line in the O x y coordinate system, which is given by the equation x - 2 3 = y - 3 1 . The point M 1 (2 , 3) belongs to the straight line, the vector a → (3 , 1) is the direction vector of this straight line.
The canonical straight line equation of the form x - x 1 a x = y - y 1 a y can be used in cases where a x or a y is zero. The presence of zero in the denominator makes the notation x - x 1 a x = y - y 1 a y conditional. The equation can be written as follows a y (x - x 1) = a x (y - y 1) .
In the case when a x \u003d 0, the canonical equation of a straight line takes the form x - x 1 0 \u003d y - y 1 a y and sets a straight line that is parallel to the ordinate axis or coincides with this axis.
The canonical equation of a straight line, provided that a y \u003d 0, takes the form x - x 1 a x \u003d y - y 1 0. Such an equation defines a straight line parallel to the x-axis or coinciding with it.
More material on the topic of the canonical equation of a straight line, see here. In the article, we provide a number of solutions to problems, as well as numerous examples that allow you to better master the topic.
Parametric equations of a straight line on a plane
These equations have the form x \u003d x 1 + a x λ y \u003d y 1 + a y λ, where x 1, y 1, a x, a y are some real numbers, of which a x and a y cannot be equal to zero at the same time. An additional parameter λ is introduced into the formula, which can take any real values.
The purpose of the parametric equation is to establish an implicit relationship between the coordinates of the points of a straight line. For this, the parameter λ is introduced.
The numbers x , y are the coordinates of some point on the line. They are calculated by parametric equations of a straight line for some real value of the parameter λ.
Example 5
Let's assume that λ = 0 .
Then x \u003d x 1 + a x 0 y \u003d y 1 + a y 0 ⇔ x \u003d x 1 y \u003d y 1, i.e., the point with coordinates (x 1, y 1) belongs to the line.
We draw your attention to the fact that the coefficients a x and a y with the parameter λ in this type of equations are the coordinates of the directing vector of the straight line.
Example 6
Consider parametric straight line equations of the form x = 2 + 3 · λ y = 3 + λ . The straight line given by the equations in the Cartesian coordinate system passes through the point (x 1 , y 1) and has a directing vector a → = (3 , 1) .
For more information, see the article "Parametric equations of a straight line on a plane".
The normal equation of a straight line has the form, A x + B y + C = 0 , where the numbers A, B, and C are such that the length of the vector n → = (A , B) is equal to one, and C ≤ 0 .
The normal vector of the line, given by the normal equation of the straight line in the rectangular coordinate system O x y, is the vector n → = (A , B) . This line passes at a distance C from the origin in the direction of the vector n → = (A , B) .
Another way to write the normal equation of a straight line is cos α x + cos β y - p = 0, where cos α and cos β are two real numbers that are direction cosines of the unit length normal vector of a straight line. This means that n → = (cos α , cos β) , the equality n → = cos 2 α + cos 2 β = 1 is true, the value p ≥ 0 and is equal to the distance from the origin to the straight line.
Example 7
Consider the general equation of the straight line - 1 2 · x + 3 2 · y - 3 = 0 . This general equation of the line is the normal equation of the line, since n → = A 2 + B 2 = - 1 2 2 + 3 2 = 1 and C = - 3 ≤ 0 .
The equation defines a straight line in the Cartesian coordinate system 0xy, the normal vector of which has coordinates - 1 2 , 3 2 . The line is removed from the origin by 3 units in the direction of the normal vector n → = - 1 2 , 3 2 .
We draw your attention to the fact that the normal equation of a straight line on a plane allows you to find the distance from a point to a straight line on a plane.
If in the general equation of the line A x + B y + C \u003d 0 the numbers A, B and C are such that the equation A x + B y + C \u003d 0 is not a normal equation of the line, then it can be reduced to a normal form. Read more about this in the article "Normal Equation of a Line".
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Consider a relation of the form F(x, y)=0 linking the variables x And at. Equality (1) will be called equation with two variables x, y, if this equality is not true for all pairs of numbers X And at. Equation examples: 2x + 3y \u003d 0, x 2 + y 2 - 25 \u003d 0,
sin x + sin y - 1 = 0.
If (1) is true for all pairs of numbers x and y, then it is called identity. Identity examples: (x + y) 2 - x 2 - 2xy - y 2 \u003d 0, (x + y) (x - y) - x 2 + y 2 \u003d 0.
Equation (1) will be called the equation of the set of points (x; y), if this equation is satisfied by the coordinates X And at any point of the set and do not satisfy the coordinates of any point that do not belong to this set.
An important concept in analytic geometry is the concept of the equation of a line. Let a rectangular coordinate system and some line α.
Definition. Equation (1) is called the line equation α
(in the created coordinate system), if this equation is satisfied by the coordinates X And at any point on the line α
, and do not satisfy the coordinates of any point that does not lie on this line.
If (1) is the line equation α, then we will say that equation (1) determines (sets) line α.
Line α can be determined not only by an equation of the form (1), but also by an equation of the form
F(P, φ) = 0, containing polar coordinates.
- equation of a straight line with a slope;
Let some straight line, not perpendicular to the axis, be given OH. Let's call tilt angle given line to the axis OH corner α by which to rotate the axis OH so that the positive direction coincides with one of the directions of the straight line. The tangent of the angle of inclination of a straight line to the axis OH called slope factor this straight line and denoted by the letter TO.
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We derive the equation of this straight line, if we know its TO and the value in the segment OV, which she cuts off on the axis OU.
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Equation (2) is called equation of a straight line with a slope. If K=0, then the line is parallel to the axis OH and its equation is y = b.
- equation of a straight line passing through two points;
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. This is the equation if y 1 ≠ y 2, can be written as: If y 1 = y 2, then the equation of the desired straight line has the form y = y 1. In this case, the line is parallel to the axis OH. If x 1 = x 2, then the line passing through the points M 1 And M 2, parallel to the axis OU, its equation has the form x = x 1.
- equation of a straight line passing through a given point with a given slope;
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and, conversely, equation (5) for arbitrary coefficients A, B, C (A And B ≠ 0 simultaneously) defines some line in a rectangular coordinate system Ohu.
Proof.
Let us first prove the first assertion. If the line is not perpendicular Oh, then it is determined by the equation of the first degree: y = kx + b, i.e. equation of the form (5), where
A=k, B=-1 And C = b. If the line is perpendicular Oh, then all its points have the same abscissa equal to the value α segment cut off by a straight line on the axis Oh.
The equation of this line has the form x = α, those. is also a first-degree equation of the form (5), where A \u003d 1, B \u003d 0, C \u003d - α. This proves the first assertion.
Let's prove converse statement. Let equation (5) be given, and at least one of the coefficients A And B ≠ 0.
If B ≠ 0, then (5) can be written as . sloping 
, we get the equation y = kx + b, i.e. an equation of the form (2) that defines a straight line.
If B = 0, That A ≠ 0 and (5) takes the form . Denoting through α, we get
x = α, i.e. equation of a straight line perpendicular Ox.
Lines defined in a rectangular coordinate system by an equation of the first degree are called first order lines.
Type equation Ah + Wu + C = 0 is incomplete, i.e. one of the coefficients is equal to zero.
1) C = 0; Ah + Wu = 0 and defines a line passing through the origin.
2) B = 0 (A ≠ 0); the equation Ax + C = 0 OU.
3) A = 0 (B ≠ 0); Wu + C = 0 and defines a line parallel Oh.
Equation (6) is called the equation of a straight line "in segments". Numbers A And b are the values of the segments that the straight line cuts off on the coordinate axes. This form of the equation is convenient for the geometric construction of a straight line.
- normal equation of a straight line;
Аx + Вy + С = 0 is the general equation of some straight line, and (5) x cos α + y sin α – p = 0(7)
its normal equation.
Since equations (5) and (7) define the same straight line, then ( A 1x + B 1y + C 1 \u003d 0 And
A 2x + B 2y + C 2 = 0 => 
) the coefficients of these equations are proportional. This means that by multiplying all the terms of equation (5) by some factor M, we obtain the equation MA x + MB y + MS = 0, coinciding with equation (7) i.e.
MA = cos α, MB = sin α, MC = - P(8)
To find the M factor, we square the first two of these equalities and add:
M 2 (A 2 + B 2) \u003d cos 2 α + sin 2 α \u003d 1
(9)