Equation of a circle on the coordinate plane

Definition 1. Number axis ( number line, coordinate line) Ox is the straight line on which point O is selected origin (origin of coordinates)(Fig.1), direction

O → x

listed as positive direction and a segment is marked, the length of which is taken to be unit of length.

Definition 2. A segment whose length is taken as a unit of length is called scale.

Each point on the number axis has a coordinate that is a real number. The coordinate of point O is zero. The coordinate of an arbitrary point A lying on the ray Ox is equal to the length of the segment OA. The coordinate of an arbitrary point A of the numerical axis that does not lie on the ray Ox is negative, and in absolute value is equal to the length of the segment OA.

Definition 3. Rectangular Cartesian coordinate system Oxy on the plane call two mutually perpendicular numerical axes Ox and Oy with the same scale And common beginning countdown at point O, and such that the rotation from ray Ox at an angle of 90° to ray Oy is carried out in the direction counterclockwise(Fig. 2).

Note. The rectangular Cartesian coordinate system Oxy, shown in Figure 2, is called right coordinate system, Unlike left coordinate systems, in which the rotation of the beam Ox at an angle of 90° to the beam Oy is carried out in a clockwise direction. In this guide we we consider only right-handed coordinate systems, without specifically specifying it.

If we introduce some system of rectangular Cartesian coordinates Oxy on the plane, then each point of the plane will acquire two coordinates – abscissa And ordinate, which are calculated as follows. Let A be an arbitrary point on the plane. Let us drop perpendiculars from point A A.A. 1 and A.A. 2 to straight lines Ox and Oy, respectively (Fig. 3).

Definition 4. The abscissa of point A is the coordinate of the point A 1 on the number axis Ox, the ordinate of point A is the coordinate of the point A 2 on the number axis Oy.

Designation Coordinates (abscissa and ordinate) of the point A in the rectangular Cartesian coordinate system Oxy (Fig. 4) is usually denoted A(x;y) or A = (x; y).

Note. Point O, called origin, has coordinates O(0 ; 0) .

Definition 5. In the rectangular Cartesian coordinate system Oxy, the numerical axis Ox is called the abscissa axis, and the numerical axis Oy is called the ordinate axis (Fig. 5).

Definition 6. Each rectangular Cartesian coordinate system divides the plane into 4 quarters (quadrants), the numbering of which is shown in Figure 5.

Definition 7. The plane on which a rectangular Cartesian coordinate system is given is called coordinate plane.

Note. The abscissa axis is specified on the coordinate plane by the equation y= 0, the ordinate axis is given on the coordinate plane by the equation x = 0.

Statement 1. Distance between two points coordinate plane

A 1 (x 1 ;y 1) And A 2 (x 2 ;y 2)

calculated according to the formula

Proof . Consider Figure 6.

|A 1 A 2 | 2 = = (x 2 -x 1) 2 + (y 2 -y 1) 2 .

(1)

Hence,

Q.E.D.

Equation of a circle on the coordinate plane

Let us consider on the coordinate plane Oxy (Fig. 7) a circle of radius R with center at the point A 0 (x 0 ;y 0) .

Polar coordinates

The number is called polar radius dots or first polar coordinate. Distance cannot be negative, so the polar radius of any point is . The first polar coordinate is also denoted by a Greek letter (“rho”), but I am used to the Latin version and will use it in the future.

The number is called polar angle given point or second polar coordinate. The polar angle typically varies within (the so-called principal angle values). However, it is quite acceptable to use the range, and in some cases there is a direct need to consider all angle values ​​from zero to “plus infinity”. By the way, I recommend getting used to the radian measure of an angle, since operating with degrees in higher mathematics is considered not comme il faut.

The couple is called polar coordinates dots They are easy to find and specific values. Tangent acute angle right triangle - is the ratio of the opposite side to the adjacent side: therefore, the angle itself: . According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the legs: , which means the polar radius:

Thus, .

One penguin is good, but a flock is better:


Negatively oriented corners I marked it with arrows just in case, in case some of the readers did not yet know about this orientation. If desired, you can “screw” 1 turn (rad. or 360 degrees) to each of them and get, by the way, comfortable table values:

But the disadvantage of these "traditionally" oriented angles is that they are "twisted" too far (more than 180 degrees) counterclockwise. I anticipate the question: “why is there a disadvantage and why are some negative angles needed at all?” In mathematics, the shortest and most rational paths are valued. Well, from the point of view of physics, the direction of rotation is often of fundamental importance - each of us tried to open the door by pulling the handle in the wrong direction =)

The order and technique of constructing points in polar coordinates

Beautiful pictures are beautiful, but constructing them in the polar coordinate system is a rather painstaking task. There are no difficulties with points whose polar angles are , in our example these are points ; Values ​​that are multiples of 45 degrees also do not cause much trouble: . But how to correctly and competently construct, say, a point?

You will need a checkered piece of paper, a pencil and the following drawing tools: ruler, compass, protractor. As a last resort, you can get by with just one ruler, or even... without it at all! Read on and you will get another proof that this country is invincible =)

Example 1

Construct a point in the polar coordinate system.

First of all, you need to find out the degree measure of the angle. If the corner is unfamiliar or you have doubts, then it is always better to use table or a general formula for converting radians to degrees. So our angle is (or).

Let's draw a polar coordinate system (see the beginning of the lesson) and pick up a protractor. Owners of a round instrument will have no difficulty marking 240 degrees, but most likely you will have a semicircular version of the device on your hands. The problem of the complete absence of a protractor in the presence of a printer and scissors solved by handicraft.

There are two ways: turn the sheet over and mark 120 degrees, or “screw” half a turn and look at the opposite angle. Let's choose the adult method and make a mark of 60 degrees:


Either a Lilliputian protractor, or a giant cage =) However, to measure an angle, the scale is not important.

Using a pencil, draw a thin straight line passing through the pole and the mark made:


We've sorted out the angle, now the polar radius is next. Take a compass and along the line we set its solution to 3 units, most often this is, of course, centimeters:

Now carefully place the needle on the pole, and rotational movement We make a small serif (red color). The required point was constructed:


You can do without a compass by applying the ruler directly to the constructed straight line and measuring 3 centimeters. But, as we will see later, in problems involving construction in a polar coordinate system a typical situation is when you need to mark two or large quantity points with the same polar radius, so it is more efficient to harden the metal. In particular, in our drawing, by rotating the leg of the compass 180 degrees, it is easy to make a second notch and construct a point symmetrical relative to the pole. Let’s use it to work through the material in the next paragraph:

Relationship between rectangular and polar coordinate systems

Obviously let's add to the polar coordinate system, a “regular” coordinate grid and draw a point in the drawing:

It is always useful to keep this connection in mind when drawing in polar coordinates. Although, willy-nilly, it suggests itself without any further hint.

Let's establish the relationship between polar and Cartesian coordinates using the example of a specific point. Let's consider right triangle, in which the hypotenuse is equal to the polar radius: , and the legs are equal to the “X” and “Y” coordinates of the point in the Cartesian coordinate system: .

The sine of an acute angle is the ratio of the opposite side to the hypotenuse:

The cosine of an acute angle is the ratio of the adjacent leg to the hypotenuse:

At the same time, we repeated the definitions of sine, cosine (and a little earlier tangent) from the 9th grade curriculum of a comprehensive school.

Please add working formulas in your reference book that express the Cartesian coordinates of a point through its polar coordinates - we will have to deal with them more than once, and next time right now =)

Let's find the coordinates of a point in a rectangular coordinate system:

Thus:

The resulting formulas open another loophole in the construction problem, when you can do without a protractor at all: first we find the Cartesian coordinates of the point (of course, in the draft), then we mentally find the desired place on the drawing and mark this point. On final stage draw a thin straight line that passes through the constructed point and the pole. As a result, it turns out that the angle was allegedly measured with a protractor.

It's funny that very desperate students can even do without a ruler, using instead the smooth edge of a textbook, notebook or grade book - after all, the notebook manufacturers took care of the metrics, 1 square = 5 millimeters.

All this reminded me of a well-known joke in which resourceful pilots plotted a course along a pack of Belomor =) Although, jokes aside, the joke is not so far from reality, I remember that on one of the domestic flights in the Russian Federation, all the navigation instruments in the airliner failed, and the crew successfully I landed the plane using a regular glass of water, which showed the angle of the plane relative to the ground. And the airstrip - here it is, from windshield visible

Using the Pythagorean theorem cited at the beginning of the lesson, it is easy to obtain the inverse formulas: , therefore:

The angle “phi” itself is standardly expressed through the arctangent - absolutely the same as complex number argument with all its troubles.

It is also advisable to place the second group of formulas in your reference luggage.

After detailed analysis flights with individual points, let’s move on to the natural continuation of the topic:

Equation of a line in polar coordinates

Essentially, the equation of a line in a polar coordinate system is function of polar radius from polar angle (argument). In this case, the polar angle is taken into account in radians(!) And continuously takes values ​​from to (sometimes it should be considered to infinity, or in a number of problems for convenience from to). Each value of the angle “phi” that is included in domain function, corresponds to a single value of the polar radius.

The polar function can be compared to a kind of radar - when a beam of light emanating from a pole rotates counterclockwise and “detects” (draws) a line.

A standard example of a polar curve is Archimedean spiral. The following picture shows her first round– when the polar radius following the polar angle takes values ​​from 0 to:

Further, crossing the polar axis at point , the spiral will continue to unwind, moving infinitely far from the pole. But such cases are quite rare in practice; a more typical situation is when at all subsequent revolutions we “walk along the same line” that was obtained in the range.

In the first example we come across the concept domain of definition polar function: since the polar radius is non-negative, negative angles cannot be considered here.

! Note : in some cases it is customary to use generalized polar coordinates, where the radius can be negative, and we will briefly study this approach a little later

In addition to the Archimedes spiral, there are many other famous curves, but, as they say, you can’t get enough of art, so I selected examples that are very often found in real practical tasks.

First, the simplest equations and simplest lines:

An equation of the form specifies the one emanating from the pole Ray. Indeed, think about it, if the angle value Always(whatever the “er” is) constantly, then what line is it?

Note : in the generalized polar coordinate system, this equation defines a straight line passing through the pole

An equation of the form determines... guess the first time - if for anyone Angle "phi" radius remains constant? In fact this is the definition circle centered at the pole of radius .

For example, . For clarity, let's find the equation of this line in a rectangular coordinate system. Using the formula obtained in the previous paragraph, we make the replacement:

Let's square both sides:

– equation of a circle with center at the origin of radius 2, which is what needed to be checked.

Since the creation and release of the article about linear dependence and linear independence of vectors I received several letters from site visitors who asked a question in the spirit of: “there is a simple and convenient rectangular coordinate system, why do we need another oblique affine case?” The answer is simple: mathematics strives to embrace everything and everyone! In addition, in a given situation, convenience is important - as you can see, it is much more profitable to work with a circle in polar coordinates due to the extreme simplicity of the equation.

And sometimes mathematical model anticipates scientific discoveries. So, at one time the rector of Kazan University N.I. Lobachevsky strictly proved, through an arbitrary point of the plane one can draw infinitely many straight lines, parallel to this one. As a result, he was defamed by everything scientific world, but... refute this fact no one could. Only a good century later, astronomers discovered that light in space travels along curved trajectories, where Lobachevsky’s non-Euclidean geometry, formally developed by him long before this discovery, begins to work. It is assumed that this is a property of space itself, the curvature of which is invisible to us due to small (by astronomical standards) distances.

Let's consider more meaningful construction tasks:

Example 2

Build a line

Solution: First of all, let's find domain. Since the polar radius is non-negative, the inequality must hold. Can you remember school rules solutions to trigonometric inequalities, but in simple cases like this, I recommend a faster and more visual solution:

Imagine a cosine graph. If it has not yet registered in your memory, then find it on the page Graphs of elementary functions. What does inequality tell us? It tells us that the cosine graph should be located not less abscissa axis. And this happens on the segment. And, accordingly, the interval is not suitable.

Thus, the domain of definition of our function is: , that is, the graph is located to the right of the pole (in the terminology of the Cartesian system - in the right half-plane).

In polar coordinates, there is often a vague idea of ​​​​which line defines a particular equation, so in order to construct it, you need to find the points that belong to it - and the more, the better. Usually they are limited to a dozen or two (or even less). The easiest way, of course, is to take table angle values. For greater clarity, I will “screw” one turn to negative values:

Due to the parity of the cosine the corresponding positive values ​​do not need to be counted again:

Let us depict a polar coordinate system and plot the found points, while same values It is convenient to put off the “er” at a time, making paired notches with a compass using the technology discussed above:

In principle, the line is clearly drawn, but in order to completely confirm the guess, let’s find its equation in the Cartesian coordinate system. You can apply the recently derived formulas , but I'll tell you about a more cunning trick. We artificially multiply both sides of the equation by “er”: and use more compact transition formulas:

Selecting a complete square, we bring the equation of the line to a recognizable form:

– equation of a circle with center at point , radius 2.

Since according to the condition it was simply necessary to carry out the construction and that’s it, we smoothly connect the found points with a line:

Ready. It's okay if it turns out a little uneven, you didn't have to know that it was a circle ;-)

Why didn't we consider the angle values ​​outside the interval? The answer is simple: there is no point. Due to the periodicity of the function, we are faced with an endless run along the constructed circle.

It is easy to carry out a simple analysis and come to the conclusion that an equation of the form specifies a circle of diameter with a center at point . Figuratively speaking, all such circles “sit” on the polar axis and necessarily pass through the pole. If then funny company will migrate to the left - to the continuation of the polar axis (think about why).

Similar task for independent decision:

Example 3

Construct a line and find its equation in a rectangular coordinate system.

Let us systematize the procedure for solving the problem:

First of all, we find the domain of definition of the function; for this it is convenient to look at sinusoid to immediately understand where the sine is non-negative.

In the second step, we calculate the polar coordinates of the points using table angle values; Analyze whether it is possible to reduce the number of calculations?

In the third step, we plot the points in the polar coordinate system and carefully connect them with a line.

And finally, we find the equation of the line in the Cartesian coordinate system.

A sample solution is at the end of the lesson.

General algorithm and we detail the construction technique in polar coordinates
and significantly speed up in the second part of the lecture, but before that we’ll get acquainted with another common line:

Polar Rose

That's right, we're talking about a flower with petals:

Example 4

Construct lines given by equations in polar coordinates

There are two approaches to constructing a polar rose. First, let's follow the knurled track, assuming that the polar radius cannot be negative:

Solution:

a) Let’s find the domain of definition of the function:

This trigonometric inequality is also easy to solve graphically: from the materials of the article Geometric transformations of graphs it is known that if the argument of a function is doubled, then its graph will shrink to the ordinate axis by 2 times. Please find the graph of the function in the first example of this lesson. Where is this sinusoid located above the x-axis? At intervals . Consequently, the inequality is satisfied by the corresponding segments, and domain our function: .

Generally speaking, the solution to the inequalities under consideration is a union of an infinite number of segments, but, again, we are interested in only one period.

Perhaps some readers will find it easier to use an analytical method for finding the domain of definition; I will call it “slicing a round pie.” We'll cut into equal parts and, first of all, find the boundaries of the first piece. We reason as follows: sine is non-negative, When his argument ranges from 0 to rad. inclusive. In our example: . Dividing all parts of the double inequality by 2, we obtain the required interval:

Now we begin to sequentially “cut equal pieces of 90 degrees” counterclockwise:

– the found segment is, of course, included in the domain of definition;

– next interval – not included;

– next segment – ​​included;

– and finally, the interval – is not included.

Just like a daisy - “loves, doesn’t love, loves, doesn’t love” =) With the difference that there is no fortune telling here. Yes, it’s just some kind of love in the Chinese way….

So, and the line represents a rose with two identical petals. It is quite acceptable to draw the drawing schematically, but it is highly advisable to correctly find and mark tops of petals. The vertices correspond to midpoints of segments of the domain of definition, which in this example have obvious angular coordinates . Wherein petal lengths are:

Here is the natural result of a caring gardener:

It should be noted that the length of the petal can be easily seen from the equation - since the sine is limited: , then the maximum value of “er” will certainly not exceed two.

b) Let's build a line, given by the equation. Obviously, the length of the petal of this rose is also two, but, first of all, we are interested in the domain of definition. Applicable analytical method"cuts": sine is non-negative when its argument is in the range from zero to “pi” inclusive, in in this case: . We divide all parts of the inequality by 3 and get the first interval:

Next, we begin “cutting the pie into pieces” by rad. (60 degrees):
– the segment will enter the definition domain;
– interval – will not be included;
– segment – ​​will fit;
– interval – will not be included;
– segment – ​​will fit;
– interval – will not be included.

The process is successfully completed at 360 degrees.

Thus, the scope of definition is: .

The actions carried out in whole or in part are easy to carry out mentally.

Construction. If in the previous paragraph everything worked out well with right angles and angles of 45 degrees, then here you will have to tinker a little. Let's find tops of petals. Their length was visible from the very beginning of the task; all that remains is to calculate the angular coordinates, which are equal to the midpoints of the segments of the definition domain:

Please note that there must be equal spaces between the tops of the petals, in this case 120 degrees.

It is advisable to mark the drawing into 60-degree sectors (delimited green lines) and draw the directions of the vertices of the petals (gray lines). It is convenient to mark the vertices themselves using a compass - measure a distance of 2 units once and make three notches in the drawn directions of 30, 150 and 270 degrees:

Ready. I understand that this is a troublesome task, but if you want to arrange everything wisely, you will have to spend time.

Let us formulate a general formula: an equation of the form , is a natural number), defines a polar -petaled rose, the petal length of which is equal to .

For example, the equation specifies a quatrefoil with a petal length of 5 units, the equation specifies a 5-petal rose with a petal length of 3 units. etc.

A rectangular coordinate system on a plane is formed by two mutually perpendicular coordinate axes X’X and Y’Y. The coordinate axes intersect at point O, which is called the origin, a positive direction is selected on each axis. The positive direction of the axes (in a right-handed coordinate system) is chosen so that when the X'X axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the Y'Y axis. The four angles (I, II, III, IV) formed by the coordinate axes X'X and Y'Y are called coordinate angles (see Fig. 1).

The position of point A on the plane is determined by two coordinates x and y. The x coordinate is equal to the length of the segment OB, the y coordinate is equal to the length of the segment OC in the selected units of measurement. Segments OB and OC are defined by lines drawn from point A parallel to the Y’Y and X’X axes, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A. It is written as follows: A(x, y).

If point A lies in coordinate angle I, then point A has a positive abscissa and ordinate. If point A lies in coordinate angle II, then point A has a negative abscissa and a positive ordinate. If point A lies in coordinate angle III, then point A has a negative abscissa and ordinate. If point A lies in coordinate angle IV, then point A has a positive abscissa and a negative ordinate.

Rectangular coordinate system in space is formed by three mutually perpendicular coordinate axes OX, OY and OZ. The coordinate axes intersect at point O, which is called the origin, on each axis a positive direction is selected, indicated by arrows, and a unit of measurement for the segments on the axes. The units of measurement are the same for all axes. OX - abscissa axis, OY - ordinate axis, OZ - applicate axis. The positive direction of the axes is chosen so that when the OX axis is rotated counterclockwise by 90°, its positive direction coincides with the positive direction of the OY axis, if this rotation is observed from the positive direction of the OZ axis. Such a coordinate system is called right-handed. If thumb right hand take the X direction as the X direction, the index one as the Y direction, and the middle one as the Z direction, then a right-handed coordinate system is formed. Similar fingers of the left hand form the left coordinate system. It is impossible to combine the right and left coordinate systems so that the corresponding axes coincide (see Fig. 2).

The position of point A in space is determined by three coordinates x, y and z. The x coordinate is equal to the length of the segment OB, the y coordinate is the length of the segment OC, the z coordinate is the length of the segment OD in the selected units of measurement. The segments OB, OC and OD are defined by planes drawn from point A parallel to the planes YOZ, XOZ and XOY, respectively. The x coordinate is called the abscissa of point A, the y coordinate is called the ordinate of point A, the z coordinate is called the applicate of point A. It is written as follows: A(a, b, c).

Orty

A rectangular coordinate system (of any dimension) is also described by a set of unit vectors aligned with the coordinate axes. The number of unit vectors is equal to the dimension of the coordinate system and they are all perpendicular to each other.

In the three-dimensional case, such unit vectors are usually denoted i j k or e x e y e z. In this case, in the case of a right-handed coordinate system, the following formulas with the vector product of vectors are valid:

• [i j]=k ;
• [j k]=i ;
• [k i]=j .

Story

The rectangular coordinate system was first introduced by Rene Descartes in his work “Discourse on Method” in 1637. Therefore, the rectangular coordinate system is also called - Cartesian coordinate system. The coordinate method of describing geometric objects marked the beginning of analytical geometry. Pierre Fermat also contributed to the development of the coordinate method, but his works were first published after his death. Descartes and Fermat used the coordinate method only on the plane.

The coordinate method for three-dimensional space was first used by Leonhard Euler already in the 18th century.

see also

Links

Wikimedia Foundation. 2010.

• Cartesian coordinate system
• Cartesian degree

See what “Cartesian coordinates” are in other dictionaries:

CARTESINE COORDINATES- (Cartesian coordinate system) a coordinate system on a plane or in space, usually with mutually perpendicular axes and equal scales along the axes; rectangular Cartesian coordinates. Named after R. Descartes... Big Encyclopedic Dictionary

Cartesian coordinates- A coordinate system consisting of two perpendicular axes. The position of a point in such a system is formed using two numbers that determine the distance from the coordinate center along each of the axes. Informational topics... ... Technical Translator's Guide

Cartesian coordinates- (Cartesian coordinate system), a coordinate system on a plane or in space, usually with mutually perpendicular axes and equal scales along the axes; rectangular Cartesian coordinates. Named after R. Descartes... encyclopedic Dictionary

Cartesian coordinates- Dekarto koordinatės statusas T sritis Standartizacija ir metrologija apibrėžtis Tiesinė plokštumos arba erdvės koordinačių sistema. Joje ašių masteliai paprastai būna lygūs. atitikmenys: engl. Cartesian coordinates vok. kartesische Koordinaten, f… Penkiakalbis aiškinamasis metrologijos terminų žodynas

Cartesian coordinates- Dekarto koordinatės statusas T sritis fizika atitikmenys: engl. Cartesian coordinates; grid coordinates vok. kartesische Koordinaten, f rus. Cartesian coordinates, f pranc. coordonnées cartésiennes, f … Fizikos terminų žodynas

CARTESINE COORDINATES- a method of determining the position of points on a plane by their distances to two fixed perpendicular straight axes. This concept is already seen in Archimedes and Appologis of Perga more than two thousand years ago and even among the ancient Egyptians. For the first time this... ... Mathematical Encyclopedia

CARTESINE COORDINATES- Cartesian coordinate system [named after the French. philosopher and mathematician R. Descartes (R. Descartes; 1596 1650)], a coordinate system on a plane or in space, usually with mutually perpendicular axes and equal scales along the axes rectangular D ... Big Encyclopedic Polytechnic Dictionary

CARTESINE COORDINATES- (Cartesian coordinate system), a coordinate system on a plane or in space, usually with mutually perpendicular axes and equal scales along the rectangular axes. Named after R. Descartes... Natural science. encyclopedic Dictionary

CARTESINE COORDINATES- The system for positioning any point found on the bones relative to two axes intersecting at right angles. Developed by René Descartes, this system became the basis for standard methods graphical presentation of data. Horizontal line… … Dictionary in psychology

Coordinates- Coordinates. On the plane (left) and in space (right). COORDINATES (from the Latin co together and ordinatus ordered), numbers that determine the position of a point on a straight line, plane, surface, in space. Coordinates are distances... Illustrated Encyclopedic Dictionary

Instructions

Write down mathematical operations in text form and enter them into the search query field at home page Google site if you cannot use the calculator, but have access to the Internet. This search engine has a built-in multifunctional calculator, which is much easier to use than any other. There is no interface with buttons - all data must be entered in text form in a single field. For example, if known coordinates extreme points segment in a three-dimensional coordinate system A(51.34 17.2 13.02) and A(-11.82 7.46 33.5), then coordinates midpoint segment C((51.34-11.82)/2 (17.2+7.46)/2 (13.02+33.5)/2). By entering (51.34-11.82)/2 into the search query field, then (17.2+7.46)/2 and (13.02+33.5)/2, you can use Google to get coordinates C(19.76 12.33 23.26).

The standard equation of a circle allows you to find out several important information about this figure, for example, the coordinates of its center, the length of the radius. In some problems, on the contrary, you need to create an equation using given parameters.

Instructions

Determine what information you have about the circle based on the task given to you. Remember that the ultimate goal is to determine the coordinates of the center as well as the diameter. All your actions should be aimed at achieving this particular result.

Use data on the presence of points of intersection with coordinate lines or other lines. Please note that if the circle passes through the abscissa axis, the second one will have coordinate 0, and if it passes through the ordinate axis, then the first one. These coordinates will allow you to find the coordinates of the circle's center and also calculate the radius.

Don't forget about the basic properties of secants and tangents. In particular, the most useful theorem is that at the point of contact the radius and tangent form a right angle. But please note that you may be asked to prove all theorems used during the course.

Solve the most standard types to learn to immediately see how to use certain data for the equation of a circle. So, in addition to the already mentioned tasks with directly given coordinates and those in which information about the presence of intersection points is given, to compile the equation of a circle, you can use knowledge about the center of the circle, the length of the chord and on which this chord lies.

To solve, construct an isosceles triangle, the base of which will be the given chord, and the equal sides will be the radii. Compile from which you can easily find the necessary data. To do this, it is enough to use the formula for finding the length of a segment in a plane.

Video on the topic

A circle is understood as a figure that consists of many points on a plane equidistant from its center. Distance from center to points circle called radius.

An ordered system of two or three intersecting axes perpendicular to each other with a common origin (origin of coordinates) and a common unit of length is called rectangular Cartesian coordinate system .

General Cartesian coordinate system (affine coordinate system) may not necessarily include perpendicular axes. In honor of the French mathematician Rene Descartes (1596-1662), just such a coordinate system is named in which a common unit of length is measured on all axes and the axes are straight.

Rectangular Cartesian coordinate system on a plane has two axes and rectangular Cartesian coordinate system in space - three axes. Each point on a plane or in space is defined by an ordered set of coordinates - numbers corresponding to the unit of length of the coordinate system.

Note that, as follows from the definition, there is a Cartesian coordinate system on a straight line, that is, in one dimension. The introduction of Cartesian coordinates on a line is one of the ways by which any point on a line is associated with a well-defined real number, that is, a coordinate.

The coordinate method, which arose in the works of Rene Descartes, marked a revolutionary restructuring of all mathematics. It became possible to interpret algebraic equations(or inequalities) in the form of geometric images (graphs) and, conversely, look for solutions to geometric problems using analytical formulas and systems of equations. Yes, inequality z < 3 геометрически означает полупространство, лежащее ниже плоскости, параллельной координатной плоскости xOy and located above this plane by 3 units.

Using the Cartesian coordinate system, the membership of a point on a given curve corresponds to the fact that the numbers x And y satisfy some equation. Thus, the coordinates of a point on a circle with a center at a given point ( a; b) satisfy the equation (x - a)² + ( y - b)² = R² .

Rectangular Cartesian coordinate system on a plane

Two perpendicular axes on a plane with a common origin and the same scale unit form Cartesian rectangular coordinate system on the plane . One of these axes is called the axis Ox, or x-axis , the other - the axis Oy, or y-axis . These axes are also called coordinate axes. Let us denote by Mx And My respectively, the projection of an arbitrary point M on the axis Ox And Oy. How to get projections? Let's go through the point M Ox. This straight line intersects the axis Ox at the point Mx. Let's go through the point M straight line perpendicular to the axis Oy. This straight line intersects the axis Oy at the point My. This is shown in the picture below.

x And y points M we will call the values ​​of the directed segments accordingly OMx And OMy. The values ​​of these directed segments are calculated accordingly as x = x0 - 0 And y = y0 - 0 . Cartesian coordinates x And y points M abscissa And ordinate . The fact that the point M has coordinates x And y, is denoted as follows: M(x, y) .

Coordinate axes divide the plane into four quadrant , the numbering of which is shown in the figure below. It also shows the arrangement of signs for the coordinates of points depending on their location in a particular quadrant.

In addition to Cartesian rectangular coordinates on a plane, the polar coordinate system is also often considered. About the method of transition from one coordinate system to another - in the lesson polar coordinate system .

Rectangular Cartesian coordinate system in space

Cartesian coordinates in space are introduced in complete analogy with Cartesian coordinates in the plane.

Three mutually perpendicular axes in space (coordinate axes) with a common origin O and with the same scale unit they form Cartesian rectangular coordinate system in space .

One of these axes is called an axis Ox, or x-axis , the other - the axis Oy, or y-axis , the third - axis Oz, or axis applicate . Let Mx, My Mz- projections of an arbitrary point M space on the axis Ox , Oy And Oz respectively.

Let's go through the point M OxOx at the point Mx. Let's go through the point M plane perpendicular to the axis Oy. This plane intersects the axis Oy at the point My. Let's go through the point M plane perpendicular to the axis Oz. This plane intersects the axis Oz at the point Mz.

Cartesian rectangular coordinates x , y And z points M we will call the values ​​of the directed segments accordingly OMx, OMy And OMz. The values ​​of these directed segments are calculated accordingly as x = x0 - 0 , y = y0 - 0 And z = z0 - 0 .

Cartesian coordinates x , y And z points M are called accordingly abscissa , ordinate And applicate .

Coordinate axes taken in pairs are located in coordinate planes xOy , yOz And zOx .

Problems about points in a Cartesian coordinate system

Example 1.

A(2; -3) ;

B(3; -1) ;

C(-5; 1) .

Find the coordinates of the projections of these points onto the abscissa axis.

Solution. As follows from the theoretical part of this lesson, the projection of a point onto the abscissa axis is located on the abscissa axis itself, that is, the axis Ox, and therefore has an abscissa equal to the abscissa of the point itself, and an ordinate (coordinate on the axis Oy, which the x-axis intersects at point 0), equal to zero. So we get the following coordinates of these points on the x-axis:

Ax(2;0);

Bx(3;0);

Cx (-5; 0).

Example 2. In the Cartesian coordinate system, points are given on the plane

A(-3; 2) ;

B(-5; 1) ;

C(3; -2) .

Find the coordinates of the projections of these points onto the ordinate axis.

Solution. As follows from the theoretical part of this lesson, the projection of a point onto the ordinate axis is located on the ordinate axis itself, that is, the axis Oy, and therefore has an ordinate equal to the ordinate of the point itself, and an abscissa (coordinate on the axis Ox, which the ordinate axis intersects at point 0), which is equal to zero. So we get the following coordinates of these points on the ordinate axis:

Ay(0;2);

By(0;1);

Cy(0;-2).

Example 3. In the Cartesian coordinate system, points are given on the plane

A(2; 3) ;

B(-3; 2) ;

C(-1; -1) .

Ox .

Ox Ox Ox, will have the same abscissa as the given point, and an ordinate equal in absolute value to the ordinate of the given point, and opposite in sign. So we get the following coordinates of points symmetrical to these points relative to the axis Ox :

A"(2; -3) ;

B"(-3; -2) ;

C"(-1; 1) .

Solve problems using the Cartesian coordinate system yourself, and then look at the solutions

Example 4. Determine in which quadrants (quarters, drawing with quadrants - at the end of the paragraph “Rectangular Cartesian coordinate system on a plane”) a point can be located M(x; y) , If

1) xy > 0 ;

2) xy < 0 ;

3) x − y = 0 ;

4) x + y = 0 ;

5) x + y > 0 ;

6) x + y < 0 ;

7) x − y > 0 ;

8) x − y < 0 .

Example 5. In the Cartesian coordinate system, points are given on the plane

A(-2; 5) ;

B(3; -5) ;

C(a; b) .

Find the coordinates of points symmetrical to these points relative to the axis Oy .

Let's continue to solve problems together

Example 6. In the Cartesian coordinate system, points are given on the plane

A(-1; 2) ;

B(3; -1) ;

C(-2; -2) .

Find the coordinates of points symmetrical to these points relative to the axis Oy .

Solution. Rotate 180 degrees around the axis Oy directional segment from the axis Oy up to this point. In the figure, where the quadrants of the plane are indicated, we see that the point symmetrical to the given one relative to the axis Oy, will have the same ordinate as the given point, and an abscissa equal in absolute value to the abscissa of the given point and opposite in sign. So we get the following coordinates of points symmetrical to these points relative to the axis Oy :

A"(1; 2) ;

B"(-3; -1) ;

C"(2; -2) .

Example 7. In the Cartesian coordinate system, points are given on the plane

A(3; 3) ;

B(2; -4) ;

C(-2; 1) .

Find the coordinates of points symmetrical to these points relative to the origin.

Solution. We rotate the directed segment going from the origin to the given point by 180 degrees around the origin. In the figure, where the quadrants of the plane are indicated, we see that a point symmetrical to the given point relative to the origin of coordinates will have an abscissa and ordinate equal in absolute value to the abscissa and ordinate of the given point, but opposite in sign. So we get the following coordinates of points symmetrical to these points relative to the origin:

A"(-3; -3) ;

B"(-2; 4) ;

C(2; -1) .

Example 8.

A(4; 3; 5) ;

B(-3; 2; 1) ;

C(2; -3; 0) .

Find the coordinates of the projections of these points:

1) on a plane Oxy ;

2) on a plane Oxz ;

3) to the plane Oyz ;

4) on the abscissa axis;

5) on the ordinate axis;

6) on the applicate axis.

1) Projection of a point onto a plane Oxy is located on this plane itself, and therefore has an abscissa and ordinate equal to the abscissa and ordinate of a given point, and an applicate equal to zero. So we get the following coordinates of the projections of these points onto Oxy :

Axy (4; 3; 0);

Bxy (-3; 2; 0);

Cxy(2;-3;0).

2) Projection of a point onto a plane Oxz is located on this plane itself, and therefore has an abscissa and applicate equal to the abscissa and applicate of a given point, and an ordinate equal to zero. So we get the following coordinates of the projections of these points onto Oxz :

Axz (4; 0; 5);

Bxz (-3; 0; 1);

Cxz (2; 0; 0).

3) Projection of a point onto a plane Oyz is located on this plane itself, and therefore has an ordinate and applicate equal to the ordinate and applicate of a given point, and an abscissa equal to zero. So we get the following coordinates of the projections of these points onto Oyz :

Ayz(0; 3; 5);

Byz (0; 2; 1);

Cyz (0; -3; 0).

4) As follows from the theoretical part of this lesson, the projection of a point onto the abscissa axis is located on the abscissa axis itself, that is, the axis Ox, and therefore has an abscissa equal to the abscissa of the point itself, and the ordinate and applicate of the projection are equal to zero (since the ordinate and applicate axes intersect the abscissa at point 0). We obtain the following coordinates of the projections of these points onto the abscissa axis:

Ax(4;0;0);

Bx (-3; 0; 0);

Cx(2;0;0).

5) The projection of a point onto the ordinate axis is located on the ordinate axis itself, that is, the axis Oy, and therefore has an ordinate equal to the ordinate of the point itself, and the abscissa and applicate of the projection are equal to zero (since the abscissa and applicate axes intersect the ordinate axis at point 0). We obtain the following coordinates of the projections of these points onto the ordinate axis:

Ay(0; 3; 0);

By (0; 2; 0);

Cy(0;-3;0).

6) The projection of a point onto the applicate axis is located on the applicate axis itself, that is, the axis Oz, and therefore has an applicate equal to the applicate of the point itself, and the abscissa and ordinate of the projection are equal to zero (since the abscissa and ordinate axes intersect the applicate axis at point 0). We obtain the following coordinates of the projections of these points onto the applicate axis:

Az (0; 0; 5);

Bz (0; 0; 1);

Cz(0; 0; 0).

Example 9. In the Cartesian coordinate system, points are given in space

A(2; 3; 1) ;

B(5; -3; 2) ;

C(-3; 2; -1) .

Find the coordinates of points symmetrical to these points with respect to:

1) plane Oxy ;

2) planes Oxz ;

3) planes Oyz ;

4) abscissa axes;

5) ordinate axes;

6) applicate axes;

7) origin of coordinates.

1) “Move” the point on the other side of the axis Oxy Oxy, will have an abscissa and ordinate equal to the abscissa and ordinate of a given point, and an applicate equal in magnitude to the aplicate of a given point, but opposite in sign. So, we get the following coordinates of points symmetrical to the data relative to the plane Oxy :

A"(2; 3; -1) ;

B"(5; -3; -2) ;

C"(-3; 2; 1) .

2) “Move” the point on the other side of the axis Oxz to the same distance. From the figure displaying the coordinate space, we see that a point symmetrical to a given one relative to the axis Oxz, will have an abscissa and applicate equal to the abscissa and applicate of a given point, and an ordinate equal in magnitude to the ordinate of a given point, but opposite in sign. So, we get the following coordinates of points symmetrical to the data relative to the plane Oxz :

A"(2; -3; 1) ;

B"(5; 3; 2) ;

C"(-3; -2; -1) .

3) “Move” the point on the other side of the axis Oyz to the same distance. From the figure displaying the coordinate space, we see that a point symmetrical to a given one relative to the axis Oyz, will have an ordinate and an aplicate equal to the ordinate and an aplicate of a given point, and an abscissa equal in value to the abscissa of a given point, but opposite in sign. So, we get the following coordinates of points symmetrical to the data relative to the plane Oyz :

A"(-2; 3; 1) ;

B"(-5; -3; 2) ;

C"(3; 2; -1) .

By analogy with symmetrical points on a plane and points in space that are symmetrical to data relative to planes, we note that in the case of symmetry with respect to some axis of the Cartesian coordinate system in space, the coordinate on the axis with respect to which the symmetry is given will retain its sign, and the coordinates on the other two axes will be the same in absolute value as the coordinates of a given point, but opposite in sign.

4) The abscissa will retain its sign, but the ordinate and applicate will change signs. So, we obtain the following coordinates of points symmetrical to the data relative to the abscissa axis:

A"(2; -3; -1) ;

B"(5; 3; -2) ;

C"(-3; -2; 1) .

5) The ordinate will retain its sign, but the abscissa and applicate will change signs. So, we obtain the following coordinates of points symmetrical to the data relative to the ordinate axis:

A"(-2; 3; -1) ;

B"(-5; -3; -2) ;

C"(3; 2; 1) .

6) The applicate will retain its sign, but the abscissa and ordinate will change signs. So, we obtain the following coordinates of points symmetrical to the data relative to the applicate axis:

A"(-2; -3; 1) ;

B"(-5; 3; 2) ;

C"(3; -2; -1) .

7) By analogy with symmetry in the case of points on a plane, in the case of symmetry about the origin of coordinates, all coordinates of a point symmetrical to a given one will be equal in absolute value to the coordinates of a given point, but opposite to them in sign. So, we obtain the following coordinates of points symmetrical to the data relative to the origin.