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Question 139738: please help me with this problem. It is from a mathe work sheet. Thnak you.
Find the center and the radius of the circle.
4.) x2 + 12x + 36 + (y-4)2 = 49
Answer by JSmall(7)   (Show Source):
You can put this solution on YOUR website! The most useful form of the equation of a circle is:
( x - h )^2 + ( y - k )^2 = r^2
When the equation of a circle is written this way, the center of the circle can be easily read: (h, k). And the radius, r, can also be read easily.
So to solve your problem we need to use Algebra to change your original equation:
x^2 + 12x + 36 + ( y - 4 )^2 = 49
to one that looks like:
( x - h )^2 + ( y - k )^2 = r^2
As I hope you can see, your original equation already has "y" in the proper form. All we need to do is change
x^2 + 12x + 36
to something like
( x - h )^2
and change whatever number is on the right to something like
r^2
To change
x^2 + 12x + 36
to something like
( x - h )^2
we do something called "completing the square".
For
x^2 + 12x + 36
completing the square happens to be very easy.
x^2 + 12x + 36
is equal to
( x + 6 )^2
(If this is not obvious, multiply ( x + 6 ) ( x + 6 ) [Most people use "FOIL" for this.] and see that
x^2 + 12x + 36 = ( x + 6 )^2
Now ( x + 6 )^2 is not quite in the form ( x - h )^2. We need to rewrite it as a subtraction: ( x + 6 )^2 = ( x - (-6))^2
The only thing remaining is to rewrite the 49 on the right side as something squared. Since 49 = 7^2 this is pretty simple.
So our final version of the equation is
( x - (-6))^2 + ( y - 4 )^2 = 7^2
The center of the circle is (-6, 4) and the radius is 7.
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