Lesson A MUST-READ LESSON:Deriving the general equation of any Circle

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First, let us start with the most basic circle

CIRCLE WITH CENTER AT (0,0):

The equation of this circle is

_____________________________________{{{x^2+y^2=r^2}}}______________________________________________

where r= a parameter, equivalent to the length of the radius of the circle

{{{drawing( 700, 700, 
          -5, 5,
          -5, 5,
          grid(1),
          red( circle( 0, 0, 3 ) ),
          locate( 0, 0, X )
 )}}}

You can see a circle with aradius of 3 and the center is at (0,0)

Looking at it, you can now write the equation of the circle.

_____________________________________{{{x^2+y^2=9}}}________________________________________________

NOTE: 9 is obtained from squaring th e radius----{{{3^2=9}}}

Something different happens if the center is not at the origin.

CIRCLES WITH CENTER AT (h,k):

You have seen a circle w/ a radius of 3 and center at (0,0). If we transfer the center to another place, it would not be the same.

To find the equation of the circle with the radius r at (h,k), let us see the following formula:
__________________________________{{{(x-h)^2+(y-k)^2=r^2}}}_________________________________________

Look at the different of 'version' of our circle with a radius of 3! Look below:
{{{drawing( 700, 700, 
          -7, 7,
          -7, 7,
          grid(1),
          red( circle( 3, -2, 3 ) ),
          locate( 3, -2, X )
 )}}}

The center is mark with an x and found to be (3,-2)!
Knowing that the center of any circle is "formatted" to be (h,k), you will know that h=3 and -2!

Thus, the equation of our new circle is,
______________________________{{{(x-3)^2+(y-(-2))^2=3^2}}}__________________________________________

_________________________________{{{(x-3)^2+(y+2)^2=9}}}____________________________________________

Now, let's try to simplify the equation {{{(x-h)^2+(y-k)^2=r^2}}}

{{{(x-h)^2+(y-k)^2=r^2}}}
{{{((x-h)(x-h))+(y-k)^2=r^2}}}
{{{(x(x-h)-h(x-h))+(y-k)^2=r^2}}}
{{{((x^2-hx)-(hx-h^2))+(y-k)^2=r^2}}}
{{{(x^2-hx-hx+h^2)+(y-k)^2=r^2}}}
{{{(x^2-2hx+h^2)+(y-k)^2=r^2}}}
{{{(x^2-2hx+h^2)+((y-k)(y-k))=r^2}}}
{{{(x^2-2hx+h^2)+(y(y-k)-k(y-k))=r^2}}}
{{{(x^2-2hx+h^2)+((y^2-ky)-(ky-k^2))=r^2}}}
{{{(x^2-2hx+h^2)+(y^2-ky-ky+k^2)=r^2}}}
{{{(x^2-2hx+h^2)+(y^2-2ky+k^2)=r^2}}}
{{{x^2-2hx+h^2+y^2-2ky+k^2=r^2}}}
{{{x^2+(-2hx)+h^2+y^2+(-2ky)+k^2=r^2}}}
{{{x^2+y^2+(-2hx)+(-2ky)+h^2+k^2=r^2}}}
{{{x^2+y^2-2hx-2ky+h^2+k^2=r^2}}}
{{{x^2+y^2-2hx-2ky+h^2+k^2-r^2=0}}}

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