Let a Cartesian rectangular coordinate system Oxy and some line L be given on the plane .

Definition. The equation F(x;y)=0 (1) called line equationL(with respect to a given coordinate system) if this equation satisfies the x and y coordinates of any point lying on the line L, and does not satisfy the x and y coordinates of any point not lying on the line L.

That. line on the plane is the locus of points (M(x;y)) whose coordinates satisfy equation (1).

Equation (1) defines the line L.

Example. Circle equation.

Circle- a set of points equidistant from a given point M 0 (x 0, y 0).

Point M 0 (x 0, y 0) - circle center.

For any point M(x; y) lying on the circle, the distance MM 0 =R (R=const)

MM 0 ==R

(x-x 0 ) 2 +(y-y 0 ) 2 =R 2 –(2) – the equation of a circle of radius R centered at the point M 0 (x 0, y 0).

Parametric line equation.

Let the x and y coordinates of the points of the line L be expressed using the parameter t:

(3) - parametric equation of the line in DSC

where the functions (t) and (t) are continuous with respect to the parameter t (in a certain range of variation of this parameter).

Eliminating the parameter t from equation (3), we obtain equation (1).

Let us consider the line L as a path traveled by a material point, continuously moving according to a certain law. Let the variable t represent the time counted from some initial moment. Then the task of the law of motion is the task of the x and y coordinates of the moving point as some continuous functions x=(t) and y=(t) of time t.

Example. Let us derive a parametric equation for a circle of radius r>0 centered at the origin. Let M(x, y) be an arbitrary point of this circle, and t be the angle between the radius vector and the Ox axis, counted counterclockwise.

Then x=r cos x y=r sin t. (4)

Equations (4) are parametric equations of the considered circle. The parameter t can take any value, but in order for the point M(x, y) to go around the circle once, the parameter change area is limited to the half-segment 0t2.

Squaring and adding equations (4), we obtain the general equation of the circle (2).

2. Polar coordinate system (psc).

Let us choose the axis L on the plane ( polar axis) and determine the point of this axis О ( pole). Any point of the plane is uniquely defined by polar coordinates ρ and φ, where

ρ – polar radius, equal to the distance from the point M to the pole O (ρ≥0);

φ – corner between vector direction OM and the L axis ( polar angle). M(ρ ; φ )

Line equation in UCS can be written:

ρ=f(φ) (5) explicit line equation in PCS

F=(ρ; φ) (6) implicit line equation in PCS

Relationship between Cartesian and polar coordinates of a point.

(x; y) (ρ ; φ ) From triangle OMA:

tg φ=(restoration of the angleφ according to the well-knowntangent is producedtaking into account in which quadrant the point M is located).(ρ ; φ )(x; y). x=ρcos φ,y= ρsin φ

Example . Find the polar coordinates of the points M(3;4) and P(1;-1).

For M:=5, φ=arctg (4/3). For P: ρ=; φ=Π+arctg(-1)=3Π/4.

Classification of flat lines.

Definition 1. The line is called algebraic, if in some Cartesian rectangular coordinate system, if it is defined by the equation F(x;y)=0 (1), in which the function F(x;y) is an algebraic polynomial.

Definition 2. Any non-algebraic line is called transcendent.

Definition 3. The algebraic line is called line of ordern, if in some Cartesian rectangular coordinate system this line is determined by equation (1), in which the function F(x;y) is an algebraic polynomial of the nth degree.

Thus, a line of the nth order is a line defined in some Cartesian rectangular system by an algebraic equation of degree n with two unknowns.

The following theorem helps establish the correctness of Definitions 1,2,3.

Theorem(documentation on p. 107). If a line in some Cartesian rectangular coordinate system is determined by an algebraic equation of degree n, then this line in any other Cartesian rectangular coordinate system is determined by an algebraic equation of the same degree n.

The equation of a line as a locus of points. Various types of equations of a straight line. Study of the general equation of a straight line. Construction of a straight line according to its equation

Line equation is called an equation with variables x And y, which is satisfied by the coordinates of any point of 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 formulate the 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 by equality the common property of all points M lines;
3) the segments (and angles) included in this equality are expressed in terms of 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 given in one of the following forms:

1. Equation of a straight line with a slope

y = kx + b, (1)

Where k- the slope 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 value of the segment cut off by a straight line on the y-axis. At b= 0 equation (1) has the form y = kx and the corresponding line passes through the origin.

Equation (1) can be used to define any straight line in the plane that is not perpendicular to the axis Ox.

Equation of a straight line with a slope is allowed relative to the current coordinate y.

2. General equation of a straight line

Ax + By + C = 0. (2)

Particular 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 any coordinate system. Coordinate systems can be different depending on the choice of basis and origin.

Definition. The line equation is the ratio y \u003d 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.

2. Equation of a straight line on a plane

Definition. Any straight line in the plane can be given by the 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 a straight line.

IN Depending on the values ​​of the constants A, B and C, the following special cases are possible:

- the line passes through the origin

C \u003d 0, A ≠ 0, B ≠ 0 ( By + C \u003d 0) - the line is parallel to the Ox axis

B = 0, A ≠ 0, C ≠ 0( Ax + C = 0) - the line is parallel to the Oy axis

B = C = 0, A ≠ 0 - the line coincides with the Oy axis

A = C = 0, B ≠ 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.

3. Equation of a straight line with respect to a point and a 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 a straight line passing through the point А(1,2) perpendicular to the vector n (3, − 1) .

Compose for A=3 and B=-1 the equation of a straight line: 3x − y + C = 0 . To find the coefficient

With we substitute the coordinates of the given point A into the resulting expression. We get: 3 − 2 + C \u003d 0, therefore C \u003d -1.

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

4. Equation of a straight 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 a straight line,

passing through these points:	x − x1				y − y1				z−z1
			− x				− y				− z

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

On the plane, the straight line equation 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 straight line. x2 − x1

5. Equation of a straight line in terms of a point and a slope

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

is called the equation of a straight line with slope k.

6. Equation of a straight line by 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 assignment of a straight line through a point and a directing vector of a 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 the directing vector of the line

Ax + By + C = 0 .

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

We will look for the equation of the desired straight line in the form: Ax + By + C = 0 . According to the definition, the coefficients must satisfy the following conditions: 1A + (− 1) B = 0 , i.e. A=B. Then the straight line equation looks like: Ax + Ay + C = 0 , or x + y + C / A = 0 . at x=1, y=2 we get C/A=-3, i.e. desired equation: x + y − 3 = 0

7. Equation of a straight line in segments

If in the general equation of the line Ax + By + C \u003d 0, C ≠ 0, then, dividing by -С,

we get: −

x−

y = 1 or

1, where a = −

b = −

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 is the coordinate of the point of intersection of the line with the Oy axis.

8. Normal equation of a straight line

is called a normalizing factor, then we obtain x cosϕ + y sinϕ − p = 0, the normal equation of a straight 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 lines on a plane

Definition. If two lines are given y \u003d k 1 x + b 1, y \u003d 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. The straight line passing through the point M1 (x1, y1) and perpendicular to the straight line y \u003d kx + b is represented by the equation:

y − y = −

(x − x )

10. Distance from point to line

If a point M(x0, y0) is given, then the distance to the line Ax + By + C = 0

defined as d =

Ax0 + By0 + C

Example. Determine the angle between the lines: y = − 3x + 7, y = 2x + 1.

k = − 3, k

2tg ϕ =

2 − (− 3)

1;ϕ = π / 4.

1− (− 3)2

Example. Show,

that the lines 3 x − 5 y + 7 = 0 and 10 x + 6 y − 3 = 0

are perpendicular.

We find: k 1 \u003d 3/ 5, k 2 \u003d - 5 / 3, k 1 k 2 \u003d - 1, therefore, the lines are perpendicular.

Example. Given the vertices of the triangle A(0 ; 1) , B (6 ; 5) , C (1 2 ; - 1) .

Find the equation for the height drawn from vertex C.
We find the equation of the side AB:	x − 0		y − 1					y − 1		; 4x = 6y − 6
	6 − 0		5 − 1

2x − 3y + 3 = 0; y = 2 3 x + 1.

The desired height equation has the form: Ax + By + C = 0 or y = kx + bk = − 3 2 Then

y = − 3 2 x + b . Because height passes through point C, then its coordinates satisfy this equation: − 1 = − 3 2 12 + b , whence b=17. Total: y = − 3 2 x + 17 .

Answer: 3x + 2y - 34 = 0 .

Target: Consider the concept of a line on a plane, give examples. Based on the definition of a line, introduce the concept of the equation of a straight line in a plane. Consider the types of a straight line, give examples and ways to set a straight line. To consolidate the ability to translate the equation of a straight line from a general form into an equation of a straight line “in segments”, with a slope.

1. Equation of a line on a plane.
2. Equation of a straight line on a plane. Types of equations.
3. Ways to set a straight line.

1. Let x and y be two arbitrary variables.

Definition: A relation of the form F(x,y)=0 is called equation , if it is not valid for any pairs of numbers x and y.

Example: 2x + 7y - 1 \u003d 0, x 2 + y 2 - 25 \u003d 0.

If the equality F(x,y)=0 holds for any x, y, then, consequently, F(x,y) = 0 is an identity.

Example: (x + y) 2 - x 2 - 2xy - y 2 = 0

They say x is 0 and y is 0 satisfy the equation , if, when they are substituted into this equation, it turns into a true equality.

The most important concept of analytic geometry is the concept of the equation of a line.

Definition: The equation of a given line is the equation F(x,y)=0, which is satisfied by the coordinates of all points lying on this line, and not satisfied by the coordinates of any of the points not lying on this line.

The line defined by the equation y = f(x) is called the graph of the function f(x). Variables x and y are called current coordinates, because they are coordinates of a variable point.

Some examples line definitions.

1) x - y \u003d 0 \u003d\u003e x \u003d y. This equation defines a straight line:

2) x 2 - y 2 \u003d 0 => (x-y) (x + y) \u003d 0 => points must satisfy either the equation x - y \u003d 0, or the equation x + y \u003d 0, which corresponds to a pair of intersecting lines that are bisectors of coordinate angles:

3) x 2 + y 2 \u003d 0. Only one point O (0,0) satisfies this equation.

2. 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 2 ¹ 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 \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 with 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 equation of a straight line with slope k.

Equation of a straight line in segments.

If in the general equation of the straight line Ax + Vy + С = 0 С ¹ 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 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 sides of the equation Ax + Wy + C = 0 divided by a number called normalizing factor, then we get

xcosj + ysinj - p = 0 is the 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.

3. Equation of a straight line by a point and a slope.

Let the slope of the straight line be equal to k, the straight line passes through the point M(x 0, y 0). Then the equation of the straight line is found by the formula: y - y 0 \u003d k (x - x 0)

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.

Fraction = k is called slope factor straight.

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ANALYTIC GEOMETRY

Lecture No. 7. Topic 1 : Lines in the plane and their equations

1.1. Lines and their equations in Cartesian coordinates

In analytic geometry, lines on a plane are considered as the locus of points (g.m.t.) that have the same property common to all points of the line.

Definition. Line equation
is an equation with two variablesX And at, which is satisfied by the coordinates of any point on the line and is not satisfied by the coordinates of any other point that does not lie on this line.

The converse is also true, i.e. any equationat

of the form , generally speaking, in the Cartesian

coordinate system (DSC) defines a line

as a H.M.T., whose coordinates satisfy

this equation. ABOUT X

Remark 1. Not every type equation defines a line. For example, for the equation
there are no points, coordinates, which would satisfy this equation. Such cases will not be considered further.This is the case of so-called imaginary lines.

P example 1.Write an equation for a circle with a radiusR centered on a point
.

For any point lyingatM

on a circle, by definitionR

circles as g.m.t., equidistant

from the point , we get the equationX

1.2. Parametric equations of lines

There is another way to define a line on a plane using equations calledparametric:

Example 1 The line is given by parametric equations

It is required to obtain the equation of this line in DSC.

Exclude the parametert . To do this, we square both sides of these equations and add

Example 2 The line is given by parametric equations

A

It is required to obtain the equation

this line in DSC. —a a

Let's do the same, then we get

— A

Remark 2. It should be noted that the parametert in mechanics is time.

1.3. Line equation in polar coordinates

DSC is not the only way to determine the position of a point, and therefore the equation of a line. On a plane, it is often expedient to use the so-called polar coordinate system (PSC).

P SC will be determined by specifying a point O - pole and beam OR emanating from this point, which is called the polar axis. Then the position of any point is determined by two numbers: the polar radius
and polar angle is the angle between

polar axis and polar radius.

Positive reference direction

polar angle from the polar axis

counted counterclockwise.

For all points of the plane
, O R

and for the uniqueness of the polar angle it is considered
.

If the beginning of the DSC is combined with

pole, and the O axis X send by

polar axis, it is easy to verifyat

in connection between polar and

Cartesian coordinates:

ABOUT X R

Back,

(1)

If the line equation in DSC has the form , then in PSC - Then from this equation you can get an equation in the form

Example 3 Write the equation of the circle in UCS if the center of the circle is at the pole.

Using transition formulas (1) from DSC to PSC, we obtain

P example 4.Write an equation for a circle

if the pole is on the circle and the polar axisat

passes through the diameter.

Let's do the same

About 2 R X

R

This equation can also be obtained

from geometric representations (see fig.).

P example 5.Plot Line

Let's move on to PSC. The equation

will take the form
ABOUT

We will plot the line withA

taking into account its symmetry and ODZ

features:

This line is calledlemniscate Bernoulli.

1.4. Coordinate system transformation.

Line equation in new coordinate system

1. Parallel transfer of DSC.at

Consider two DSCs havingM

the same direction of the axes, but

different origins.

In the coordinate system ABOUT hu dot

regarding the system
ABOUT X

has coordinates
. Then we have

And

In coordinate form, the resulting vector equality has the form

or
. (2)

Formulas (2) are formulas for the transition from the "old" coordinate system ABOUT huto the "new" coordinate system and vice versa.

Example 5 Obtain the equation of a circle by performing a parallel translation of the coordinate systemto the center of the circle.

AND from formulas (2) it follows
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