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

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This Lesson (A MUST-READ LESSON:Deriving the general equation of any Circle) was created by by HyperBrain(694) About Me : View Source, Show
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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%5E2%2By%5E2=r%5E2______________________________________________

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

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%5E2%2By%5E2=9________________________________________________

NOTE: 9 is obtained from squaring th e radius----3%5E2=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:
__________________________________%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2_________________________________________
Look at the different of 'version' of our circle with a radius of 3! Look below:


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,
______________________________%28x-3%29%5E2%2B%28y-%28-2%29%29%5E2=3%5E2__________________________________________
_________________________________%28x-3%29%5E2%2B%28y%2B2%29%5E2=9____________________________________________

Now, let's try to simplify the equation %28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2
%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2
%28%28x-h%29%28x-h%29%29%2B%28y-k%29%5E2=r%5E2
%28x%28x-h%29-h%28x-h%29%29%2B%28y-k%29%5E2=r%5E2
%28%28x%5E2-hx%29-%28hx-h%5E2%29%29%2B%28y-k%29%5E2=r%5E2
%28x%5E2-hx-hx%2Bh%5E2%29%2B%28y-k%29%5E2=r%5E2
%28x%5E2-2hx%2Bh%5E2%29%2B%28y-k%29%5E2=r%5E2
%28x%5E2-2hx%2Bh%5E2%29%2B%28%28y-k%29%28y-k%29%29=r%5E2
%28x%5E2-2hx%2Bh%5E2%29%2B%28y%28y-k%29-k%28y-k%29%29=r%5E2
%28x%5E2-2hx%2Bh%5E2%29%2B%28%28y%5E2-ky%29-%28ky-k%5E2%29%29=r%5E2
%28x%5E2-2hx%2Bh%5E2%29%2B%28y%5E2-ky-ky%2Bk%5E2%29=r%5E2
%28x%5E2-2hx%2Bh%5E2%29%2B%28y%5E2-2ky%2Bk%5E2%29=r%5E2
x%5E2-2hx%2Bh%5E2%2By%5E2-2ky%2Bk%5E2=r%5E2
x%5E2%2B%28-2hx%29%2Bh%5E2%2By%5E2%2B%28-2ky%29%2Bk%5E2=r%5E2
x%5E2%2By%5E2%2B%28-2hx%29%2B%28-2ky%29%2Bh%5E2%2Bk%5E2=r%5E2
x%5E2%2By%5E2-2hx-2ky%2Bh%5E2%2Bk%5E2=r%5E2
x%5E2%2By%5E2-2hx-2ky%2Bh%5E2%2Bk%5E2-r%5E2=0

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