A straight line on a plane and in space.
The study of the properties of geometric shapes using algebra is called analytical geometry , and we will use the so-called coordinate method .
A line on a plane is usually defined as a set of points that have their own properties. The fact that the coordinates (numbers) x and y of a point lying on this line are analytically written as some equation.
Def.1 Line equation (curve equation) on the Oxy plane is called the equation (*), which is satisfied by the x and y coordinates of each point of the given line and not satisfied by the coordinates of any other point not lying on this line.
From Definition 1 it follows that any line on the plane corresponds to some equation between the current coordinates ( x,y ) points of this line and vice versa, to any equation there corresponds, generally speaking, some line.
This gives rise to two main problems of analytic geometry in the plane.
1. A line is given in the form of a set of points. You need to write an equation for this line.
2. Given the equation of the line. It is necessary to study its geometric properties (shape and location).
Example. Do the points lie A(-2;1) And IN (1;1) on line 2 X +at +3=0?
The problem of finding the intersection points of two lines given by the equations and is reduced to finding the coordinates that satisfy the equation of both lines, i.e. to the solution of a system of two equations in two unknowns.
If this system has no real solutions, then the lines do not intersect.
The concept of a line is introduced in a similar way in the UCS.
A line on a plane can be defined by two equations
Where X And at – arbitrary point coordinates M(x; y), lying on this line, and t is a variable called parameter , the parameter defines the position of the point on the plane.
For example, if , then the value of the parameter t=2 corresponds to the point (3;4) on the plane.
If the parameter is changed, then the point on the plane moves, describing the given line. This way of defining a line is called parametric, and equation (5.1) - the parametric equation of the line.
To move from parametric equations to the general equation (*), it is necessary in some way to exclude the parameter from the two equations. However, we note that such a transition is not always expedient and not always possible.
Line on the plane can be set vector equation , where t is a scalar variable parameter. Each parameter value corresponds to a specific plane vector. When changing the parameter, the end of the vector will describe some line.
vector equation in DSC corresponds to two scalar equations
(5.1), i.e. equations of projections on the coordinate axes of the vector equation of the line is its
parametric equations.
Vector equation and the parametric equations of the line have a mechanical meaning. If a point moves on a plane, then these equations are called motion equations , and the line is the trajectory of the point, while the parameter t is the time.
Conclusion: any line on the plane corresponds to an equation of the form.
In the general case, ANY EQUATION OF THE VIEW corresponds to a certain line, the properties of which are determined by this equation (the exception is that no geometric image corresponds to an equation on a plane).
Let the coordinate system on the plane be chosen.
Def. 5.1. Line equation is called such an equation of the formF(x;y) =0, which is satisfied by the coordinates of each point lying on this line, and not by the coordinates of any point not lying on it.
Type equationF(x;y )=0 are called general equation line or an implicit equation.
Thus, the line Г is the locus of points that satisfies the given equation Г=((x, y): F(x;y)=0).
The line is also called crooked.
If a rule is specified according to which a certain number u is associated with each point M of the plane (or some part of the plane), then they say that on the plane (or on a part of the plane) "the function of a point" is given; the assignment of a function is symbolically expressed by an equality of the form u=f(M). The number u associated with the point M is called the value of this function at the point M. For example, if A is a fixed point of the plane, M is an arbitrary point, then the distance from A to M is a function of the point M. In this case, f (m) \u003d AM.
Let some function u=f(M) be given and, at the same time, a coordinate system be introduced. Then an arbitrary point M is determined by the coordinates x, y. Accordingly, the value of this function at the point M is determined by the coordinates x, y, or, as they say, u=f(M) is function of two variables x and y. A function of two variables x and y is denoted by the symbol f(x; y): if f(M)=f(x;y), then the formula u=f(x; y) is called the expression of this function in the chosen coordinate system. So, in the previous example f(M)=AM; if we introduce a Cartesian rectangular coordinate system with the origin at point A, we get the expression for this function:
u=sqrt(x^2 + y^2)
PROBLEM 3688 Given a function f (x, y)=x^2–y^2–16.
Given a function f (x, y)=x^2–y^2–16. Determine the expression of this function in the new coordinate system if the coordinate axes are rotated by -45 degrees.Parametric line equations
Denote by letters x and y the coordinates of some point M; consider two functions of the argument t:
x=φ(t), y=ψ(t) (1)
When t changes, the values x and y will, generally speaking, change, therefore, the point M will move. Equalities (1) are called parametric equations lines, which is the trajectory of the point M; the argument t is named after the parameter. If the parameter t can be excluded from equalities (1), then we obtain the equation for the trajectory of the point M in the form
A line on a plane is a set of points of this plane that have certain properties, while points that do not lie on a given line do not have these properties. The line equation defines an analytically expressed relationship between the coordinates of the points lying on this line. Let this relation be given by the equation
F( x,y)=0. (2.1)
A pair of numbers satisfying (2.1) is not arbitrary: if X given, then at cannot be anything, meaning at associated with X. When it changes X changes at, and a point with coordinates ( x,y) describes this line. If the coordinates of the point M 0 ( X 0 ,at 0) satisfy equation (2.1), i.e. F( X 0 ,at 0)=0 is a true equality, then the point M 0 lies on this line. The converse is also true.
Definition. The equation of a line on a plane is an equation that is satisfied by the coordinates of any point lying on this line, and not satisfied by the coordinates of points that do not lie on this line.
If the equation of a certain line is known, then the study of the geometric properties of this line can be reduced to the study of its equation - this is one of the main ideas of analytic geometry. There are well-developed methods for studying equations. mathematical analysis, which make it easier to study the properties of lines.
When considering lines, the term is used current point lines - variable point M( x,y) moving along this line. Coordinates X And at current point are called current coordinates line points.
If from equation (2.1) it is possible to explicitly express at
through X, i.e., write equation (2.1) in the form , then the curve defined by such an equation is called schedule functions f(x).
1. An equation is given: , or . If X takes arbitrary values, then at takes values equal to X. Therefore, the line defined by this equation consists of points equidistant from the coordinate axes Ox and Oy - this is the bisector of I-III coordinate angles (straight line in Fig. 2.1).
The equation , or , determines the bisector of the II–IV coordinate angles (the straight line in Fig. 2.1).
0 x 0 x C 0 x
rice. 2.1 fig. 2.2 fig. 2.3
2. An equation is given: , where C is some constant. This equation can be written differently: . This equation is satisfied by those and only those points, ordinates at which are equal to C for any value of the abscissa X. These points lie on a straight line parallel to the Ox axis (Fig. 2.2). Similarly, the equation defines a straight line, parallel to the axis Oh (Fig. 2.3).
Not every equation of the form F( x,y)=0 defines a line on the plane: the equation is satisfied by the only point - O(0,0), and the equation is not satisfied by any point on the plane.
In the examples given, we built a line defined by this equation according to a given equation. Consider the inverse problem: to compose its equation along a given line.
3. Compose the equation of a circle centered at the point P( a,b) And
radius R .
○ A circle centered at point P and radius R is a collection of points spaced from point P at a distance R. This means that for any point M lying on the circle, MP = R, but if the point M does not lie on the circle, then MP ≠ R.. ●
An equality of the form F(x, y) = 0 is called an equation with two variables x, y, if it is not valid for any pair of numbers x, y. They say that two numbers x \u003d x 0, y \u003d y 0 satisfy some equation of the form F (x, y) \u003d 0, if when these numbers are substituted for the variables x and y in the equation, its left side vanishes.
The equation of a given line (in the assigned coordinate system) is an equation with two variables that is satisfied by the coordinates of every point lying on this line, and not satisfied by the coordinates of every point not lying on it.
In what follows, instead of the expression “given the equation of the line F(x, y) = 0”, we will often say shorter: given the line F(x, y) = 0.
If the equations of two lines F(x, y) = 0 and Ф(x, y) = 0 are given, then the joint solution of the system
F(x, y) = 0, F(x, y) = 0
gives all their points of intersection. More precisely, each pair of numbers that is a joint solution of this system determines one of the intersection points,
157. Given points *) M 1 (2; -2), M 2 (2; 2), M 3 (2; - 1), M 4 (3; -3), M 5 (5; -5), M 6 (3; -2). Determine which of the given points lie on the line defined by the equation x + y = 0, and which do not lie on it. Which line is defined by this equation? (Show it on the drawing.)
158. On the line defined by the equation x 2 + y 2 \u003d 25, find points whose abscissas are equal to the following numbers: 1) 0, 2) -3, 3) 5, 4) 7; on the same line, find points whose ordinates are equal to the following numbers: 5) 3, 6) -5, 7) -8. Which line is defined by this equation? (Show it on the drawing.)
159. Determine which lines are determined by the following equations (build them on the drawing): 1) x - y \u003d 0; 2) x + y = 0; 3) x - 2 = 0; 4)x + 3 = 0; 5) y - 5 = 0; 6) y + 2 = 0; 7) x = 0; 8) y = 0; 9) x 2 - xy \u003d 0; 10) xy + y 2 = 0; 11) x 2 - y 2 \u003d 0; 12) xy = 0; 13) 2 - 9 = 0; 14) x 2 - 8x + 15 = 0; 15) y 2 + by + 4 = 0; 16) x 2 y - 7xy + 10y = 0; 17) y - |x|; 18) x - |y|; 19) y + |x| = 0; 20) x + |y| = 0; 21) y = |x - 1|; 22) y = |x + 2|; 23) x 2 + y 2 = 16; 24) (x - 2) 2 + (y - 1) 2 \u003d 16; 25 (x + 5) 2 + (y-1) 2 = 9; 26) (x - 1) 2 + y 2 = 4; 27) x 2 + (y + 3) 2 = 1; 28) (x - 3) 2 + y 2 = 0; 29) x2 + 2y2 = 0; 30) 2x2 + 3y2 + 5 = 0; 31) (x - 2) 2 + (y + 3) 2 + 1 = 0.
160. Lines are given: l)x + y = 0; 2) x - y \u003d 0; 3) x 2 + y 2 - 36 = 0; 4) x 2 + y 2 - 2x + y \u003d 0; 5) x 2 + y 2 + 4x - 6y - 1 = 0. Determine which of them pass through the origin.
161. Lines are given: 1) x 2 + y 2 = 49; 2) (x - 3) 2 + (y + 4) 2 = 25; 3) (x + 6) 2 + (y - Z) 2 = 25; 4) (x + 5) 2 + (y - 4) 2 = 9; 5) x 2 + y 2 - 12x + 16y - 0; 6) x 2 + y 2 - 2x + 8y + 7 = 0; 7) x 2 + y 2 - 6x + 4y + 12 = 0. Find the points of their intersection: a) with the x-axis; b) with the Oy axis.
162. Find the intersection points of two lines:
1) x 2 + y 2 - 8; x - y \u003d 0;
2) x 2 + y 2 - 16x + 4y + 18 = 0; x + y = 0;
3) x 2 + y 2 - 2x + 4y - 3 = 0; x 2 + y 2 = 25;
4) x 2 + y 2 - 8y + 10y + 40 = 0; x 2 + y 2 = 4.
163. Points M 1 (l; π/3), M 2 (2; 0), M 3 (2; π/4), M 4 (√3; π/6) and M 5 (1; 2/3π) are given in the polar coordinate system. Determine which of these points lie on the line defined in polar coordinates by the equation p = 2cosΘ, and which do not lie on it. What line is determined by this equation? (Show it on the drawing.)
164. On the line defined by the equation p \u003d 3 / cosΘ, find points whose polar angles are equal to the following numbers: a) π / 3, b) - π / 3, c) 0, d) π / 6. Which line is defined by this equation? (Build it on the drawing.)
165. On the line defined by the equation p \u003d 1 / sinΘ, find points whose polar radii are equal to the following numbers: a) 1 6) 2, c) √2. Which line is defined by this equation? (Build it on the drawing.)
166. Determine which lines are determined in polar coordinates by the following equations (build them on the drawing): 1) p \u003d 5; 2) Θ = π/2; 3) Θ = - π/4; 4) р cosΘ = 2; 5) p sinΘ = 1; 6.) p = 6cosΘ; 7) p = 10 sinΘ; 8) sinΘ = 1/2; 9) sinp = 1/2.
167. Construct the following spirals of Archimedes on the drawing: 1) p = 20; 2) p = 50; 3) p = Θ/π; 4) p \u003d -Θ / π.
168. Construct the following hyperbolic spirals on the drawing: 1) p = 1/Θ; 2) p = 5/Θ; 3) р = π/Θ; 4) р= - π/Θ
169. Construct the following logarithmic spirals in the drawing: 1) p \u003d 2 Θ; 2) p = (1/2) Θ .
170. Determine the length of the segments into which the Archimedean spiral p = 3Θ cuts the beam leaving the pole and inclined to the polar axis at an angle Θ = π / 6. Make a drawing.
171. Point C is taken on the Archimedes spiral p \u003d 5 / πΘ, the polar radius of which is 47. Determine how many parts this spiral cuts the polar radius of point C. Make a drawing.
172. On a hyperbolic spiral P \u003d 6 / Θ, find a point P, the polar radius of which is 12. Make a drawing.
173. On a logarithmic spiral p \u003d 3 Θ find a point P, the polar radius of which is 81. Make a drawing.
Let's repeat * What is a quadratic equation? * What equations are called incomplete quadratic equations? * Which quadratic equation called reduced? * What is the root of a quadratic equation? * What does it mean to solve a quadratic equation? What is a quadratic equation? What equations are called incomplete quadratic equations? What quadratic equation is called reduced? What is the root of a quadratic equation? What does it mean to solve a quadratic equation? What is a quadratic equation? What equations are called incomplete quadratic equations? What quadratic equation is called reduced? What is the root of a quadratic equation? What does it mean to solve a quadratic equation?
Algorithm for solving a quadratic equation: 1. Determine which way is more rational to solve a quadratic equation 2. Choose the most rational way to solve 3. Determining the number of roots of a quadratic equation 4. Finding the roots of a quadratic equation
Additional Condition Equation Root Examples where c / a 0. b) if c / a 0, then there are no solutions even number(b \u003d 2k), and 0, to 0, c 0 ax 2 + 2kx + c \u003d 0 x 1.2 \u003d (-b ± D) / a, D 1 \u003d k 2 - ac, where k \u003d 6. The theorem is the opposite of the Vieta theorem x 2 + px + q \u003d 0x 1 + x 2 \u003d - p x 1 x 2 \u003d q
II. Special methods 7. The method of extracting the square of a binomial. Purpose: Reduce the general equation to an incomplete quadratic equation. Note: the method is applicable for any quadratic equations, but it is not always convenient to use. Used to prove the formula for the roots of a quadratic equation. Example: solve the equation x 2 -6 x + 8 = 0 8. The method of "transfer" of the senior coefficient. The roots of the quadratic equations ax 2 + bx + c = 0 and y 2 +by+ac=0 are related by the relations: and Note: the method is good for quadratic equations with "convenient" coefficients. In some cases, it allows you to solve a quadratic equation orally. Example: solve the equation 2 x 2 -9 x-5 = 0 Based on the theorems: Example: solve the equation 157 x x-177 = 0 a Example: solve the equation 203 x x+17=0 x 1 = y 1 /a, x 2 =y 2 /a
III. General methods for solving equations 11. Factoring method. Purpose: To bring a general quadratic equation to the form A(x)·B(x)=0, where A(x) and B(x) are polynomials with respect to x. Methods: Bracketing the common factor; Using abbreviated multiplication formulas; grouping method. Example: solve the equation 3 x 2 +2 x-1=0 12. Method for introducing a new variable. A good choice of a new variable makes the structure of the equation more transparent Example: solve the equation (x 2 +3 x-25) 2 -6 (x 2 +3 x-25) = - 8
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