A line on a plane can be defined using two equations

Where X And y - coordinates of an arbitrary point M(X; at), lying on this line, and t- a variable called parameter.

Parameter t determines the position of the point ( X; at) on surface.

So, if

then the parameter value t= 2 corresponds to point (4; 1) on the plane, because X = 2 + 2 = 4, y= 2 2 – 3 = 1.

If the parameter t changes, then the point on the plane moves, describing this line. This method of defining a curve is called parametric, and equations (1) - parametric line equations.

Let's consider examples of well-known curves specified in parametric form.

1) Astroid:

Where A> 0 – constant value.

At A= 2 has the form:

Fig.4. Astroid

2) Cycloid: Where A> 0 – constant.

At A= 2 has the form:

Fig.5. Cycloid

Vector equation lines

A line on a plane can be specified vector equation

Where t– scalar variable parameter.

Each parameter value t 0 corresponds to a certain plane vector. When changing a parameter t the end of the vector will describe a certain line (Fig. 6).

Vector equation of a line in a coordinate system Ohoo

correspond to two scalar equations (4), i.e. projection equations

on the coordinate axis of the vector equation of a line there are its parametric equations.

Fig.6. Vector line equation

Vector equation and parametric line equations have mechanical sense. If a point moves on a plane, then the indicated equations are called equations of motion, line – trajectory points, parameter t- time.

1. Which statement is called a corollary? Prove that a line intersecting one of two parallel lines also intersects the other. 2. Prove that

If two lines are parallel to a third line, then they are parallel.3. What theorem is called the converse of this theorem? Give examples of theorems converse to these data. 4. Prove that when two parallel lines intersect with a transversal, the angles are equal. 5. Prove that if a line is perpendicular to one of two parallel lines, then it is also perpendicular to another.6.Prove that when two parallel lines intersect with a transversal: a) the corresponding angles are equal; b) the sum of one-sided angles is 180°.

Please help me with questions on geometry (grade 9)!

2) What does it mean to decompose a vector into two

to these vectors.

9) What is the radius vector of a point? Prove that the coordinates of the point are equal to the corresponding coordinates of the vectors. 10) Derive formulas for calculating the coordinates of a vector from the coordinates of its beginning and end. 11) Derive formulas for calculating the coordinates of a vector from the coordinates of its ends. 12) Derive a formula for calculating the length of a vector from its coordinates. 13) Derive a formula for calculating the distance between two points based on their coordinates.
15) What equation is called the equation of this line? Give an example. 16) Derive the equation of a circle of a given radius with a center at a given point.
1) State and prove the lemma about collinear vectors.
3)Formulate and prove a theorem about the decomposition of a vector into two non-collinear vectors.
4) Explain how a rectangular coordinate system is introduced.
5) What are coordinate vectors?
6)Formulate and prove a statement about the decomposition of an arbitrary vector into coordinate vectors.
7) What are vector coordinates?
8) Formulate and prove the rules for finding the coordinates of the sum and difference of vectors, as well as the product of a vector and a number at given vector coordinates.
10) Derive formulas for calculating the coordinates of a vector from the coordinates of its beginning and end.
11) Derive formulas for calculating the coordinates of a vector from the coordinates of its ends. 12) Derive a formula for calculating the length of a vector from its coordinates. 13) Derive a formula for calculating the distance between two points based on their coordinates.
14) Give an example of a solution
geometric problem
using the coordinate method.
16) Derive the equation of a circle of a given radius with a center at a given point. 17) Write the equation of a circle of given radius with center at the origin. 18) Derive the equation of this line in a rectangular coordinate system. 19) Write the equation of the lines passing through this point
M0(X0:Y0) and
parallel to the axes

coordinates

20) Write the equation of the coordinate axes.

21) Give examples of using the equations of a circle and a line when solving geometric problems.
Please, I really need it! Preferably with drawings (where necessary)!
GEOMETRY 9TH GRADE.
1) State and prove the lemma about collinear vectors.
2) What does it mean to decompose a vector into two given vectors.
6)Formulate and prove a statement about the decomposition of an arbitrary vector into coordinate vectors.
7) What are vector coordinates?
8) Formulate and prove the rules for finding the coordinates of the sum and difference of vectors, as well as the product of a vector and a number at given vector coordinates.
9) What is the radius vector of a point? Prove that the coordinates of a point are equal to the corresponding coordinates of the vectors.
14) Give an example of solving a geometric problem using the coordinate method.
15)What equation is called the equation of this line? Give an example.
17) Write the equation of a circle of given radius with center at the origin.
18) Derive the equation of this line in a rectangular coordinate system.
19) Write the equation of lines passing through a given point M0 (X0: Y0) and parallel to the coordinate axes.
20) Write the equation of the coordinate axes.
parallel to the axes

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 part of the plane) “a point function is specified”; the specification of the function is symbolically expressed by an equality of the form u=f(M). The number u associated with point M is called the value of this function at point M. For example, if A is a fixed point on the plane, M is an arbitrary point, then the distance from A to M is a function of point M. B in this case f(m)=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 point M is determined by the coordinates x, y, or, as they also 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 selected 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 obtain 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 the function f (x, y)=x^2–y^2–16. Define the expression of this function in new system coordinates if the coordinate axes are rotated at an angle of –45 degrees.

Parametric line equations

Let us denote by the letters x and y the coordinates of a certain point M; Let's 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, point M will move. Equalities (1) are called parametric line equations, which is the trajectory of point M; the argument t is called a parameter. If the parameter t can be excluded from equalities (1), then we obtain the equation of the trajectory of point M in the form

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 equation of lineL(relatively given system coordinates), if this equation is satisfied by the x and y coordinates of any point lying on the line L, and is not satisfied by the x and y coordinates of any point not lying on the line L.

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

Equation (1) defines the L line.

Example. Equation of a circle.

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

Point M 0 (x 0,y 0) – center of the circle.

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

MM 0 ==R

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

Parametric equation of a line.

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

(3) – parametric equation lines in DSK

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

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

Let us consider line L as the path traversed by a material point continuously moving according to a certain law. Let the variable t represent time counted from some initial moment. Then the specification of the law of motion represents the specification of the coordinates x and y 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 with center 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 circle under consideration. The parameter t can take any value, but in order for the point M(x,y) to go around the circle once, the range of the parameter change is limited to the half-segment 0t2.

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

2. Polar coordinate system (psc).

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

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

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

Line equation in UCS can be written:

ρ=f(φ) (5) explicit equation of the line in the UCS

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

Relationship between Cartesian and polar coordinates of a point.

(x;y) (ρ ; φ ) From triangle OMA:

tan φ=(restoration of the angleφ according to the knowntangent is producedtaking into account in which quadrant point M is located).(ρ ; φ )(x;y). x=ρcosφ,y=ρsinφ

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. Every non-algebraic line is called transcendental.

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

The following theorem contributes to establishing the correctness of definitions 1,2,3.

Theorem(document 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.

An equality of the form F(x, y) = 0 is called an equation with two variables x, y if it is not true for all pairs of numbers x, y. They say that two numbers x = x 0, y = y 0 satisfy some equation of the form F(x, y) = 0 if, when substituting these numbers instead of the variables x and y into the equation, its left side becomes zero.

The equation of a given line (in a designated 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 more briefly: given the line F(x, y) = 0.

If the equations of two lines are given: F(x, y) = 0 and Ф(x, y) = 0, then the joint solution of the system

F(x,y) = 0, Ф(x, y) = 0

gives all their intersection points. 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? (Draw it on the drawing.)

158. On the line defined by the equation x 2 + y 2 = 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? (Draw it on the drawing.)

159. Determine which lines are determined by the following equations (construct them on the drawing): 1)x - y = 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 = 0; 10) xy + y 2 = 0; 11) x 2 - y 2 = 0; 12) xy = 0; 13) y 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 = 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) x 2 + 2y 2 = 0; 30) 2x 2 + 3y 2 + 5 = 0; 31) (x - 2) 2 + (y + 3) 2 + 1 = 0.

160. Given lines: l)x + y = 0; 2)x - y = 0; 3)x 2 + y 2 - 36 = 0; 4) x 2 + y 2 - 2x + y = 0; 5) x 2 + y 2 + 4x - 6y - 1 = 0. Determine which of them pass through the origin.

161. Given lines: 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 their points of intersection: a) with the Ox axis; b) with the Oy axis.

162. Find the intersection points of two lines:

1) x 2 + y 2 - 8; x - y =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. In the polar coordinate system, the 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). 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. Which line is determined by this equation? (Draw it on the drawing.)

164. On the line defined by the equation p = 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 = 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. Establish which lines are determined in polar coordinates by the following equations (construct them on the drawing): 1) p = 5; 2) Θ = π/2; 3) Θ = - π/4; 4) p 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 Archimedes spirals on the drawing: 1) p = 20; 2) p = 50; 3) p = Θ/π; 4) p = -Θ/π.

168. Construct the following hyperbolic spirals on the drawing: 1) p = 1/Θ; 2) p = 5/Θ; 3) p = π/Θ; 4) p= - π/Θ

169. Construct the following logarithmic spirals on the drawing: 1) p = 2 Θ; 2) p = (1/2) Θ.

170. Determine the lengths of the segments into which the Archimedes spiral p = 3Θ is cut by a beam emerging from the pole and inclined to the polar axis at an angle Θ = π/6. Make a drawing.

171. On the Archimedes spiral p = 5/πΘ, point C is taken, 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 = 6/Θ, find a point P whose polar radius is 12. Make a drawing.

173. On a logarithmic spiral p = 3 Θ, find a point P whose polar radius is 81. Make a drawing.