Graph of a circle

Definition:

The set of all points on a plane that are a fixed distance from a centre.

Circle on a graph :

Let us put a circle of radius 5 on a graph

Now let's check where exactly the all points are

to do so we will make a 

right-angled triangle:

And then use Pythagoras:

                               x² + y² = 5²

Like that there are infinite number of points in a circle.

Eg:- 

In all cases a point on the circle follows the rule x² + y² = radius²

Let's do a numerical on this 

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1. x value of 2, and a radius of 5

Soln--> 

Start with: x²+ y² = r²

Values we know: 2² + y²= 5²

Rearrange: y² = 5²− 2²

Square root both sides: y = ±√(5²− 2²)

Solve: y = ±√21

y ≈ ±4.58...

(Here ± means there are two possible values one with + the other with −)

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General form of Equation of a Circle

The general equation of any type of circle is represented by:

x²+ y² + 2gx + 2fy + c = 0, for all values of g, f and c.

(Adding g² + f² on both sides of the equation gives)

x² + 2gx + g²+ y² + 2fy + f²= g² + f² − c ………………(1)

Since, (x+g)²= x²+ 2gx + g² and (y+f)² =y² + 2fy + f² substituting the values in equation (1), we have

(x+g)²+ (y+f)²= g² + f²−c …………….(2)

Comparing (2) with (x−h)² + (y−k)² = a², where (h , k) is the center and ‘a’ is the radius of the circle.

h=−g, k=−f

a² = g²+ f²−c

Therefore,

x² + y² + 2gx + 2fy + c = 0, represents the circle with centre (−g,−f) and radius equal to a² = g² + f²− c.

• If g² + f² > c, then the radius of the circle is real.

• If g² + f² = c, then the radius of the circle is zero which tells us that the circle is a point that    coincides with the center. Such a type of circle is called a point circle g² + f² < c, then the radius of the circle become imaginary. Therefore, it is a circle having a real center and imaginary radius.

Polar Equation of a Circle

To find the polar form of equation of a circle, replace the value of x = r cos θ and y = r sin θ, in x² + y² = a².

Hence, we get;

(r cos θ)² + (r sin θ)² = a²

r² cos2 θ + r2 sin2 θ = a²

r² (cos²θ + sin² θ) = a²

r² (1) = a² [Using trigonometry identity]

r = a

is the polar equation of a circle with radius a and center at the origin (0,0).

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Equation Of A Circle

The standard equation of a circle is given by:

                        (x-h)² + (y-k)² = r²

Where (h,k) is the coordinates of center of the circle and r is the radius. 

Now imagine we have an equation 

                   x²+ y² + Ax + By + C = 0

And we have to convert this equation into standard form (deriving it), so 

Example: x²+ y² − 2x − 4y − 4 = 0

Soln-->

Start with: x² + y² − 2x − 4y − 4 = 0

Put xs and ys together: (x² − 2x) + (y² − 4y) − 4 = 0

Constant on right: (x² − 2x) + (y² − 4y) = 4

Now we have to complete the square for x (take half of the −2, square it, and then add it to both sides):

         (x² − 2x + (−1)²) + (y² − 4y) = 4 + (−1)²

And complete the square for y (take half of the −4, square it, and add to both sides):

(X² − 2x + (−1)²) + (y² − 4y + (−2)²) = 4 + (−1)² + (−2)²

After Simplifying it we get:

              (x² − 2x + 1) + (y² − 4y + 4) = 9

Finally:

                    (x − 1)² + (y − 2)² = 32

And we have it in Standard Form!

(Note: this used as a=1, b=2, r=3 example from before, so we got it !)