The equation provided is:

$(x + 13)^2 + (y + 2)^2 = 81$

This is the standard form of the equation of a circle, which is:

$(x - h)^2 + (y - k)^2 = r^2$

Where:

• $(h, k)$ is the center of the circle.
• $r$ is the radius of the circle.

Step-by-step breakdown:

1. In your equation, $(x + 13)$ can be rewritten as $(x - (-13))$, so the center's $x$-coordinate is $-13$.
2. Similarly, $(y + 2)$ can be rewritten as $(y - (-2))$, so the center's $y$-coordinate is $-2$.
3. The radius $r$ is the square root of 81, so $r = 9$.

Conclusion:

• The center of the circle is $(-13, -2)$.
• The radius of the circle is $9$.

Do you have any questions or need further details?

Here are 5 related questions to expand your understanding:

1. How do we derive the general form of a circle's equation?
2. What happens if the equation isn't in standard form?
3. Can the radius ever be negative?
4. How would you graph this circle on the coordinate plane?
5. What is the relationship between the distance formula and the equation of a circle?

Tip: The general form of a circle can always be rewritten to find the center and radius, even if the terms are not neatly presented.