The equation of a circle in standard form is given by:

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

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

Given:

• Center $(h, k) = (4, -5)$
• Radius $r = 3$

Substitute these values into the standard form equation:

$(x - 4)^2 + (y + 5)^2 = 3^2$

Simplify the right-hand side:

$(x - 4)^2 + (y + 5)^2 = 9$

Therefore, the equation of the circle is:

$(x - 4)^2 + (y + 5)^2 = 9$

Would you like more details or have any questions? Here are some additional questions to expand your understanding:

1. What is the general form of a circle's equation, and how can you convert from the standard form to the general form?
2. How do you determine the center and radius of a circle from its equation in general form?
3. What is the geometric meaning of the center and radius in the context of a circle?
4. How would the equation change if the circle's center was at the origin?
5. How do you find the intersection points of a circle and a line?
6. Can the radius of a circle be negative? Why or why not?
7. How would you graph the circle $(x - 4)^2 + (y + 5)^2 = 9$?
8. How does translating a circle horizontally or vertically affect its equation?

Tip: When dealing with equations of geometric shapes, always make sure to verify the center and radius from the equation to ensure correctness.