SOLUTION: What is the graph of the equation x2 + y2 - 8x + 6y + 24 = 0

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Question 153717: What is the graph of the equation x2 + y2 - 8x + 6y + 24 = 0
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

What is the graph of the equation x2 + y2 - 8x + 6y + 24 = 0
This is a circle because x%5E2 and y%5E2 have the
same coefficient when on the same side of the equation.

x%5E2+%2B+y%5E2+-+8x+%2B+6y+%2B+24+=+0

We must first get the equation in standard form:

%28x-h%29%5E2%2B%28y-k%29%5E2=+r%5E2 

Rearrange to get the x terms together and the y terms togethsr.
That is, swap the middle two terms:

x%5E2+-+8x+%2B+y%5E2+%2B+6y+%2B+24+=+0


%28x%5E2+-+8x%29+%2B+%28y%5E2+%2B+6y%29+%2B+24+=+0

Out to the side or on scratch paper,

we complete the square of

%28x%5E2-8x%29:

1. Multiply the coefficient, -8, of x, by 1%2F2,
   getting -4.
2. Then square -4, getting +16
3. Add +16 to the left side inside the first parentheses, and
   also add it to the right side:

%28x%5E2-8x%2B16%29+%2B+%28y%5E2+%2B+6y%29+%2B+24=+0%2B16   

Now we complete the square of

%28y%5E2%2B6y%29:

1. Multiply the coefficient, 6, of y, by 1%2F2,
   getting 3.
2. Then square 3, getting +9
3. Add +9 to the left side inside the first parentheses, and
   also add it to the right side:

%28x%5E2+-+8x%2B16%29+%2B+%28y%5E2+%2B+6y%2B9%29+%2B+24=+0%2B16%2B9 

Factor each parentheses:

%28x-4%29%28x-4%29+%2B+%28y%2B3%29%28y%2B3%29+%2B24=25

Write as perfect squares:

%28x-4%29%5E2+%2B+%28y%2B3%29%5E2+%2B24+=+25

Add -24 to both sides:

%28x-4%29%5E2+%2B+%28y%2B3%29%5E2+=+1

Compare to the standard form:

%28x-h%29%5E2+%2B+%28y-k%29%5E2+=+r%5E2

This has center (h,k) and radius r.

-h=-4 so h=4, -k=%2B3, so k=-3.  r%5E2=1, so r=1

so the center, (h,k) = (4,-3) and the radius is 1.

So we plot the center (4,-3).



Get a compass and open it one unit wide.  Place the sharp 
point of the compass on the center and draw the circle.
That is the graph:



Edwin