Equation of circle .

The general standard equation for the circle centered at \(\left ( h,k \right )\) with radius \( R\) is given by $$\left ( x-h \right )^2+\left ( y-k \right )^2=R^2$$

Lets see an example - We have to write the equation of circle with center 

\(\left ( 1,2\right )\) with radius \( 4\) .

The equation will be $$\left ( x-1 \right )^2+\left ( y-2 \right )^2=4^2$$

and the figure of this equation is as shown below:

Expanded form - Expanded form of circles is simply the result of expanding the squares in standard form .

For example if we want to expand the above equation of circle i.e

 $$\left ( x-1 \right )^2+\left ( y-2 \right )^2=4^2$$

                 after expending the squares we get

$$(x^2-2x+1)+(y^2-4x+4)=16$$ 

                 or we can write it as

$$x^2+y^2-2x-4y-11=0$$

Writing the standard form using the expanded equation of circles-  In this section we will learn how to write the given expended form of a circle back in standard form using "Complete square method"

lets see an example- we are given an equation of circle ,lets say

$$x^2+y^2+6x+4y-7=0$$

now we have to write this in standard form , we will use "completing the square method".

$$x^2+y^2+6x+4y=7$$

                                          or

$$(x^2+6x)+(y^2+4y)=7$$

Now apply the "Completing square method" we get 

$$(x^2+6x+9-9)+(y^2+4y+4-4)=7$$

$$(x^2+6x+9)+(y^2+4y+4)=7+9+4$$

$$(x+3)^2+(y+2)^2=20$$

               or we can write it as

$$(x-(-3))^2+(y-(-2))^2=20$$

                      or  we can write it as

$$(x-(-3))^2+(y-(-2))^2=(\sqrt{20})^2$$

so here we can see that the above equation is a standard form of circle, whit center  \(\left ( -3,-2 \right )\) and radius \(\sqrt{20}\). You can see the diagram of above circle.

Practice questions for circle (write following expanded forms in their standard form)

1).\(x^{2}+y^{2}+4x+4x=0\)

2).\(x^{2}+y^{2}+3x+6x=2\)

3).\(x^{2}+y^{2}+10x+10x=10\)

4).\(x^{2}+y^{2}=5\)

5).\(x^{2}+y^{2}+5x+3x=1\)