To convert the general form of the equation of a circle to its standard form, follow these steps:
1. Start with the given equation:
[tex]\[
3x^2 + 3y^2 + 30x - 24y - 12 = 0
\][/tex]
2. Divide every term by 3 to simplify the equation:
[tex]\[
x^2 + y^2 + 10x - 8y - 4 = 0
\][/tex]
3. Rewrite the equation grouping the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms together:
[tex]\[
x^2 + 10x + y^2 - 8y = 4
\][/tex]
4. Complete the square for the [tex]\( x \)[/tex] terms:
[tex]\[
x^2 + 10x = (x + 5)^2 - 25
\][/tex]
5. Complete the square for the [tex]\( y \)[/tex] terms:
[tex]\[
y^2 - 8y = (y - 4)^2 - 16
\][/tex]
6. Substitute these completed squares back into the equation:
[tex]\[
(x + 5)^2 - 25 + (y - 4)^2 - 16 = 4
\][/tex]
7. Simplify the equation by combining constants:
[tex]\[
(x + 5)^2 + (y - 4)^2 - 41 = 4
\][/tex]
8. Add 41 to both sides to isolate the perfect squares on the left-hand side:
[tex]\[
(x + 5)^2 + (y - 4)^2 = 45
\][/tex]
The standard form of the equation for the given circle is:
[tex]\[
(x + 5)^2 + (y - 4)^2 = 45
\][/tex]
So in the formatting desired,
[tex]\[
(x + 5)^2 + (y + (-4))^2 = 45
\][/tex]
Therefore:
[tex]\[
(x+5)^2 + (y-4)^2 = 45
\][/tex]
