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 relationship y = f(x) between the coordinates of the points that make up this line.
Note that the equation of a line can be expressed parametrically, 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, the role of the parameter is played by time.
Different types of line equation
General equation of a straight line.
Any straight line on the plane can be specified by a first-order equation
Ax + Wu + C = 0,
Moreover, the constants A and 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 line .
Depending on the values of constants A, B and C, the following special cases are possible:
C = 0, A ¹ 0, B ¹ 0 – the straight line passes through the origin
A = 0, B ¹ 0, C ¹ 0 (By + C = 0) - straight line parallel to the Ox axis
B = 0, A ¹ 0, C ¹ 0 (Ax + C = 0) – straight line parallel to the Oy axis
B = C = 0, A ¹ 0 – the straight line coincides with the Oy axis
A = C = 0, B ¹ 0 – the straight line coincides with the Ox axis
The equation of a straight line can be presented in different forms depending on any given initial conditions.
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 zero, the corresponding numerator should be set equal to zero. On the plane, the equation of the straight line written above is simplified:
if x 1 ¹ x 2 and x = x 1, if x 1 = x 2.
The fraction = k is called the slope of the line.
Equation of a straight line using a point and slope.
If the general equation of the straight line Ax + By + C = 0 is reduced 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 line Ах + Ву + 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 Ox axis, and b– the coordinate of the point of intersection of the straight line with the Oy axis.
Normal equation of a line.
If both sides of the equation Ax + By + C = 0 are divided by a number, which is called the normalizing factor, then we get
xcosj + ysinj - p = 0 –
normal equation of a 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.
The angle between straight 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 + Bу + C = 0 and A 1 x + B 1 y + C 1 = 0 are parallel when the coefficients A 1 = lA, B 1 = lB are proportional. If also С 1 = lС, 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.
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 + Bу + C = 0 is determined 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 a certain neighborhood of the point x = a (i.e., at the point x = a the function may not be defined)
Definition. A number A is called the limit of the function f(x) for x®a if for any e>0 there is a number D>0 such that for all x such that
0 < ïx - aï < D
the inequality ïf(x) - Aï is true< e.
The same definition can be written in another 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 at x ® a only at 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 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 one-sided outside the function f(x) at the point x = a. It is also said that A - final limit of a function f(x).
Equation of a line as a locus of points. Different types of straight line equations. Study of the general equation of the line. Constructing a line using its equation
Line equation called an equation with variables x And y, which is satisfied by the coordinates of any point on this line and only by them.
Variables included in the line equation x And y are called current coordinates, and literal constants are called parameters.
To create an equation of a line as a locus of points that have the same property, you need:
1) take an arbitrary (current) point M(x, y) lines;
2) write down the equality of the general property of all points M lines;
3) express the segments (and angles) included in this equality through the current coordinates of the point M(x, y) and through the data in the task.
In rectangular coordinates, the equation of a straight line on a plane is specified in one of the following forms:
1. Equation of a straight line with a slope
y = kx + b, (1)
Where k- the angular coefficient of the straight line, i.e. the tangent of the angle that the straight line forms with the positive direction of the axis Ox, and this angle is measured from the axis Ox to a straight line counterclockwise, b- the size of the segment cut off by a straight line on the ordinate axis. At b= 0 equation (1) has the form y = kx and the corresponding straight line passes through the origin.
Equation (1) can be used to define any straight line on the plane that is not perpendicular to the axis Ox.
The equation of a straight line with a slope is resolved relative to the current coordinate y.
2. General equation of a line
Ax + By + C = 0. (2)
Special cases of the general equation of a straight line.
1. Equation of a line on a plane
As you know, 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 relationship y = f (x) between the coordinates of the points that make up this line.
Note that the equation of a line can be expressed parametrically, 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, the role of the parameter is played by time.
2. Equation of a straight line on a plane
Definition. Any straight line on the plane can be specified by a first-order equation Ax + By + C = 0, and the constants A, B are not equal to zero at the same time, i.e.
A 2 + B 2 ≠ 0. This first order equation is called the general equation of the line.
IN Depending on the values of constants A, B and C, the following special cases are possible:
– a straight line passes through the origin of coordinates
C = 0, A ≠ 0, B ≠ 0( By + C = 0) - straight line parallel to the Ox axis
B = 0, A ≠ 0, C ≠ 0( Ax + C = 0) – straight line parallel to the Oy axis
B = C = 0, A ≠ 0 – the straight line coincides with the Oy axis
A = C = 0, B ≠ 0 – the straight line coincides with the Ox axis
The equation of a straight line can be presented in different forms depending on any given initial conditions.
3. Equation of a straight line from a point and normal vector
Definition. In a Cartesian rectangular coordinate system, a vector with components (A, B) is perpendicular to the 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 n (3, − 1).
With A=3 and B=-1, let’s compose the equation of the straight line: 3x − y + C = 0. To find the coefficient
Let us 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.
4. Equation of a line passing through two points
Let two points M1 (x1, y1, z1) and M2 (x2, y2, z2) be given in space, then the equation of the straight line is
passing through these points: |
x−x1 |
y−y1 |
z − z1 |
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− x |
− y |
− z |
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If any of the denominators is zero, the corresponding numerator should be set equal to zero.
On the plane, the equation of the straight line written above is simplified: y − y 1 = y 2 − y 1 (x − x 1 ) if x 2 − x 1
x 1 ≠ x 2 and x = x 1 if x 1 = x 2 .
The fraction y 2 − y 1 = k is called the slope of the line. x 2 − x 1
5. Equation of a straight line using a point and slope
If the general equation of the straight line Ax + By + C = 0 is reduced to the form:
is called the equation of a straight line with slope k.
6. 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 a (α 1 ,α 2 ) whose components satisfy the condition A α 1 + B α 2 = 0 is called a directing vector of the line
Ax + By + C = 0 .
Example. Find the equation of a straight line with a direction vector a (1,-1) and passing through the point A(1,2).
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: 1A + (− 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 get C/A=-3, i.e. required equation: x + y − 3 = 0
7. Equation of a line in segments
If in the general equation of the straight line Ax + By + C = 0, C ≠ 0, then, dividing by –C,
we get: − |
x− |
y = 1 or |
1, where a = − |
b = − |
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The geometric meaning of the coefficients is that coefficient a is the coordinate of the point of intersection of the line with the Ox axis, and b is the coordinate of the point of intersection of the line with the Oy axis.
8. Normal equation of a line
is called a normalizing factor, then we obtain x cosϕ + y sinϕ − p = 0 – the normal equation of the line.
The sign ± of the normalizing factor must be chosen so that μ C< 0 .
p is the length of the perpendicular dropped from the origin to the straight line, and ϕ is the angle formed by this perpendicular with the positive direction of the Ox axis
9. 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
Two lines are parallel if k 1 = k 2. Two lines are perpendicular if k 1 = − 1/ k 2 .
Equation of a line passing through a given point perpendicular to a given line
Definition. A straight line passing through point M1 (x1,y1) and perpendicular to the straight line y = kx + b is represented by the equation:
y − y = − |
(x − x) |
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10. Distance from a point to a line |
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If a point M(x0, y0) is given, then the distance to the straight line Ax + By + C = 0 |
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is defined as d = |
Ax0 + By0 + C |
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Example. Determine the angle between the lines: y = − 3x + 7, y = 2x + 1. |
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k = − 3, k |
2 tan ϕ = |
2 − (− 3) |
1;ϕ = π / 4. |
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1− (− 3)2 |
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Example. Show, |
that the lines 3 x − 5 y + 7 = 0 and 10 x + 6 y − 3 = 0 |
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perpendicular. |
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We find: k 1 = 3/ 5, k 2 = − 5 / 3, k 1 k 2 = − 1, therefore, the lines are perpendicular.
Example. Given are the vertices of the triangle A(0; 1), B (6; 5), C (1 2; - 1).
Find the equation of the height drawn from vertex C. |
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Find the equation of side AB: |
x − 0 |
y − 1 |
y − 1 |
; 4x = 6 y − 6 |
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6 − 0 |
5 − 1 |
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2 x − 3 y + 3 = 0; y = 2 3 x + 1.
The required height equation has the form: Ax + By + C = 0 or y = kx + bk = − 3 2 Then
y = − 3 2 x + b . Because the height passes through point C, then its coordinates satisfy this equation: − 1 = − 3 2 12 + b, from which b=17. Total: y = − 3 2 x + 17.
Answer: 3x + 2 y − 34 = 0.
Main questions of the lecture: equations of a line on a plane; various forms of the equation of a line on a plane; angle between straight lines; conditions of parallelism and perpendicularity of lines; distance from a point to a line; second-order curves: circle, ellipse, hyperbola, parabola, their equations and geometric properties; equations of a plane and a line in space.
An equation of the form is called an equation of a straight line in general form.
If we express it in this equation, then after the replacement we get an equation called the equation of a straight line with an angular coefficient, and where is the angle between the straight line and the positive direction of the abscissa axis. If in the general equation of a straight line we transfer the free coefficient to the right side and divide by it, we obtain an equation in segments
Where and are the points of intersection of the line with the abscissa and ordinate axes, respectively.
Two lines in a plane are called parallel if they do not intersect.
Lines are called perpendicular if they intersect at right angles.
Let two lines and be given.
To find the point of intersection of the lines (if they intersect), it is necessary to solve the system with these equations. The solution to this system will be the point of intersection of the lines. Let us find the conditions for the relative position of two lines.
Since, the angle between these straight lines is found by the formula
From this we can conclude that when the lines will be parallel, and when they will be perpendicular. If the lines are given in general form, then the lines are parallel under the condition and perpendicular under the condition
The distance from a point to a straight line can be found using the formula
Normal equation of a circle:
An ellipse is the geometric locus of points on a plane, the sum of the distances from which to two given points, called foci, is a constant value.
The canonical equation of an ellipse has the form:
where is the semimajor axis, is the semiminor axis and. The focal points are at the points. The vertices of an ellipse are the points. The eccentricity of an ellipse is the ratio
A hyperbola is the locus of points on a plane, the modulus of the difference in distances from which to two given points, called foci, is a constant value.
The canonical equation of a hyperbola has the form:
where is the semimajor axis, is the semiminor axis and. The focal points are at the points. The vertices of a hyperbola are the points. The eccentricity of a hyperbola is the ratio
The straight lines are called asymptotes of the hyperbola. If, then the hyperbola is called equilateral.
From the equation we obtain a pair of intersecting lines and.
A parabola is the geometric locus of points on a plane, from each of which the distance to a given point, called the focus, is equal to the distance to a given straight line, called the directrix, and is a constant value.
Canonical parabola equation
Let us consider a relation of the form F(x, y)=0, connecting variables x And at. We will call equality (1) equation with two variables x, y, if this equality is not true for all pairs of numbers X And at. Examples of equations: 2x + 3y = 0, x 2 + y 2 – 25 = 0,
sin x + sin y – 1 = 0.
If (1) is true for all pairs of numbers x and y, then it is called identity. Examples of identities: (x + y) 2 - x 2 - 2xy - y 2 = 0, (x + y)(x - y) - x 2 + y 2 = 0.
We will call equation (1) equation of a set of points (x; y), if this equation is satisfied by the coordinates X And at any point of the set and are not satisfied by the coordinates of any point that does not belong to this set.
An important concept in analytical geometry is the concept of the equation of a line. Let a rectangular coordinate system and a certain line be given on the plane 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 lying on the line α
, and do not satisfy the coordinates of any point not lying on this line.
If (1) is the equation of the line α, then we will say that equation (1) defines (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 an angular coefficient;
Let some straight line, not perpendicular, to the axis be given OH. Let's call inclination angle given straight line to the axis OH corner α , to which the axis needs to be rotated OH so that the positive direction coincides with one of the directions of the straight line. Tangent of the angle of inclination of the straight line to the axis OH called slope this line and is denoted by the letter TO.
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Let us derive the equation of this line if we know its TO and the value in the segment OB, which it cuts off on the axis OU.
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Equation (2) is called equation of a straight line with an angular coefficient. If K=0, then the straight line is parallel to the axis OH and its equation is y = b.
- equation of a 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 line has the form y = y 1. In this case, the straight line is parallel to the axis OH. If x 1 = x 2, then the straight 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 a certain straight line in a rectangular coordinate system Ooh.
Proof.
First, let's prove the first statement. 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 = 1, B = 0, C = - α. This proves the first statement.
Let us prove the 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 in the form . Flat 
, 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 by α, we get
x = α, i.e. equation of a line perpendicular Oh.
Lines defined in a rectangular coordinate system by an equation of the first degree are called first order lines.
Equation of the form Ax + Wu + C = 0 is incomplete, i.e. Some of the coefficients are equal to zero.
1) C = 0; Ah + Wu = 0 and defines a straight 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 straight 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 line;
Аx + Вy + С = 0 is the general equation of a certain 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 = 0 And
A 2x + B 2y + C 2 = 0 => 
) the coefficients of these equations are proportional. This means that by multiplying all terms of equation (5) by a certain factor M, we obtain the equation MA x + MV y + MS = 0, coinciding with equation (7) i.e.
MA = cos α, MB = sin α, MC = - P(8)
To find the factor M, we square the first two of these equalities and add:
M 2 (A 2 + B 2) = cos 2 α + sin 2 α = 1