Write the equation of the circle in standard form.
9x^2 + 9y^2 + 72x − 108y + 368 = 0
Identify its center and radius.
center (x, y) =
radius =
9x^2 + 9y^2 + 72x − 108y + 368 = 0
well put all the x stuff on the left first maybe
9x^2 + 72 x = -9 y^2 +108 y - 368
divide by 9
x^2 + 8 x = -y^2 +12y - 368/9
complete square on left by adding(8/2)^2 = 4^2 = 16 to both sides
x^2 + 8 x + 16 = -y^2 +12 y - 368/9 + 144/9
(x+4)^2 = -y^2 + 12 y - 224/9
now put the y things alone on the left
-y^2 + 12 y = (x+4)^2 + 224/9
do the complete the square thing again
y^2 - 12 y + 36 = -(x+4)^2 - 224/9 + 324/9
(y-6)^2+ (x+4)^2 = r^2 = 100/9 = (10/3)^2
To write the equation of the circle in standard form, we need to complete the square for both the x and y variables.
First, let's group the x-terms and y-terms together:
9x^2 + 9y^2 + 72x − 108y + 368 = 0
Rearranging the equation:
9x^2 + 72x + 9y^2 − 108y + 368 = 0
Now, let's complete the square for the x-terms. We divide the coefficient of x, which is 72, by 2 and square the result:
9(x^2 + 8x) + 9y^2 − 108y + 368 = 0
Adding the square of half the x-coefficient to both sides of the equation:
9(x^2 + 8x + 16) + 9y^2 − 108y + 368 + 9(16) = 0
Simplifying:
9(x + 4)^2 + 9y^2 − 108y + 432 = 0
Now, let's complete the square for the y-terms. We divide the coefficient of y, which is -108, by 2 and square the result:
9(x + 4)^2 + 9(y^2 − 12y) + 432 = 0
Adding the square of half the y-coefficient to both sides of the equation:
9(x + 4)^2 + 9(y^2 − 12y + 36) + 432 + 9(36) = 0
Simplifying:
9(x + 4)^2 + 9(y − 6)^2 + 648 = 0
Now, the equation is in standard form:
9(x + 4)^2 + 9(y − 6)^2 + 648 = 0
The center of the circle is given by the opposite of the x and y-coordinates of the completed squares, which are (-4, 6).
The radius of the circle is given by the square root of the constant term divided by the coefficient of (x + 4) and (y - 6). In this case, the square root of 648 divided by 9, which simplifies to 4√9 = 12.
Therefore, the center of the circle is (-4, 6), and the radius is 12.