To determine which equation represents the general form of a circle with a center at [tex]\((-2, -3)\)[/tex] and a diameter of 8 units, we need to follow these steps:
1. Find the radius of the circle:
- The radius [tex]\( r \)[/tex] of the circle is half of the diameter. Given the diameter is 8 units:
[tex]\[
r = \frac{8}{2} = 4
\][/tex]
2. Write the standard form of the circle equation:
- The standard form of a circle with center [tex]\((h, k)\)[/tex] and radius [tex]\(r\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
- Substituting in the center [tex]\((-2, -3)\)[/tex] and radius [tex]\(r = 4\)[/tex]:
[tex]\[
(x + 2)^2 + (y + 3)^2 = 4^2
\][/tex]
- Simplifying further:
[tex]\[
(x + 2)^2 + (y + 3)^2 = 16
\][/tex]
3. Expand the equation:
- Expanding [tex]\((x + 2)^2\)[/tex] and [tex]\((y + 3)^2\)[/tex]:
[tex]\[
(x^2 + 4x + 4) + (y^2 + 6y + 9) = 16
\][/tex]
- Combine the terms:
[tex]\[
x^2 + 4x + 4 + y^2 + 6y + 9 = 16
\][/tex]
4. Move all terms to one side of the equation to form the general equation of the circle:
[tex]\[
x^2 + y^2 + 4x + 6y + 4 + 9 - 16 = 0
\][/tex]
- Simplifying the constants:
[tex]\[
x^2 + y^2 + 4x + 6y + 13 - 16 = 0
\][/tex]
- Therefore, the general form is:
[tex]\[
x^2 + y^2 + 4x + 6y - 3 = 0
\][/tex]
Considering the answer options given:
- [tex]\(x^2 + y^2 + 4 x + 6 y - 51 = 0\)[/tex]
- [tex]\(x^2 + y^2 - 4 x - 6 y - 51 = 0\)[/tex]
- [tex]\(x^2 + y^2 + 4 x + 6 y - 3 = 0\)[/tex]
- [tex]\(x^2 + y^2 - 4 x - 6 y - 3 = 0\)[/tex]
The correct equation is:
[tex]\[
x^2 + y^2 + 4 x + 6 y - 3 = 0
\][/tex]
Thus, the equation that represents the general form of the circle is:
[tex]\[
x^2 + y^2 + 4 x + 6 y - 3 = 0
\][/tex]
