Instruction

1

Analytically the circumference is given by equation of the form (x-x0)2+(y-y0)2=R2 where x0 and y0

*are the coordinates***of the center****of the circle**, R is its radius. Thus, the center**of the circle**(x0;y0) here explicitly.2

Example. Set the center of the figure defined in the Cartesian coordinate system by the equation (x-2)2+(y-5)2=25.Solution. This equation is the equation

**of a circle**. Its center has*coordinates*(2;5). The radius of such**circle**is equal to 5.3

The equation x2+y2=R2 corresponds to a

**circle**with centre at the origin, i.e., at the point (0;0). The equation (x-x0)2+y2=R2 means that the center**of the circle**has*coordinates*(x0;0) and lies on the x-axis. The equation in the form x2+(y-y0)2=R2 says about the location**of centre**with coordinates (0;y0) on the y-axis.4

The General equation

**of a circle**in analytic geometry will be written as: x2+y2+Ax+By+C=0. To bring this equation to the abovementioned in mind, it is necessary to group members and allocate full squares: [x2+2(A/2)x+(A/2)2]+[y2+2(B/2)y+(B/2)2]+C-(A/2)2-(B/2)2=0. To highlight the full squares, as you can see, you need to add additional quantities: (A/2)2 and (B/2)2. Below the equal sign is preserved, the same value should be deducted. The addition and subtraction of the same number does not alter the equation.5

Thus: [x+(A/2)]2+[y+(B/2)]2=(A/2)2+(B/2)2. From this equation it is clear that x0=-A/2, y0=-B/2, R=√[(A/2)2+(B/2)2-C]. By the way, the expression for the radius can be simplified. Comnote both sides of R=√[(A/2)2+(B/2)2-C] 2. Then: 2R=√[A2+B2-4C]. Hence R=1/2·√[A2+B2-4C].

6

A circle cannot be the graph of a function in Cartesian coordinates, since, by definition, in a function, each x corresponds to only one y value, and

**the circumference**of these Millennials will be two. To see this, swipe perpendicular to the axis Ox intersecting the circle. You will see that the points of intersection of the two.7

But the circle can be represented as the Union of two functions: y=y0±√[R2-(x-x0)2]. Here x0 and y0, respectively, represent the desired

*coordinates***of the center****of the circle**. The coincidence**of the center****of the circle**with the origin the Union of the functions takes the form: y=√[R2-x2].Note

Two circles having the center point with the same coordinates, are called concentric. If they are given by the equations (x-x0)2+(y-y0)2=R2 and (x-x0')2+(y-y0')2=R'2, then x0=x0', y0=y0'. In the General equation for a concentric circle A1=A2 and B1=B2.

Useful advice

By the way, in physics, a circle can be considered as a thin uniform ring. The center of this ring will be the center of mass (or centre of inertia) of the body. If the ring has mass m and radius r, through centre, perpendicular to the plane of the ring to hold the axle, the moment of inertia of a ring concerning an axis is equal to mr2. The moment of inertia is fundamentally important when considering the rotational motion.

# Advice 2 : How to find the coordinates of the midpoint

The line segment is defined by two extreme points consists of multiple points lying on passing through extreme points of a straight line. If the segment is placed in any coordinate system, by finding the mid-points of its projections onto each of the axes, it is possible to know

**the coordinates****of the mid***segment*. In fact, the operation is reduced to finding the average of numbers for each of the coordinate axes.Instruction

1

Divide in half the total of the start and end coordinates of the extreme points

*of the segment*along each axis to determine**the coordinates**of the midpoint along this axis. For example, suppose the cut is placed in a three-dimensional coordinate system XYZ and the known**coordinates**of its extreme points A(Xa,Ya,Za) and C(Xc,Yc,Zc). Then**the coordinates of**its midpoint E(Xe,Ye,Ze) can be calculated by the formula Xe=(Xa+Xc)/2, Ye=(Ya+Yc)/2, Ze=(Za+Zc)/2.2

Use any of the calculators, if you average the coordinate values of the endpoints

*of a segment*in mind is not possible. If you do not have such a gadget, use a software calculator from Windows. It can be run if click "start" to open the main menu system. In the menu go to "Standard", then to "Utility" and then in the section "All programs", select "Calculator". You can do without the main menu if you press the key combination WIN + R, type calc, then press Enter.3

Summarize in pairs the start and end

**coordinates**of the extreme points*of the segment*along each axis and divide the result by two. Interface software calculator simulates an ordinary calculator, and numeric values and symbols of mathematical operations like clicking the mouse cursor on the screen, or pressing the corresponding keys on the keyboard. Any difficulties with these computations should not occur.4

Record mathematical operations in text form and enter them in the search query box on the main page of Google, if for some reason can't use a calculator, but have access to the Internet. This search engine has a built-in multi-function calculator to use which is much easier than any other. Here there is no interface with buttons to enter all data must be in text form in a single field. For example, if you know

**the coordinates**of the endpoints*of the segment*in the three-dimensional coordinate system A(51,34 17,2 of 13.02) and A(-11,82 7,46 33,5), then**the coordinates**of the midpoint*of a segment*C((51,34-11,82)/2 (17,2+7,46)/2 (13,02+33,5)/2). Typing in the search query box (51,34-11,82)/2, then (17,2+7,46)/2 and (of 13.02+33,5)/2, you can use Google to obtain**the coordinates**C(19,76 of 12.33 23,26).