**Write the equation of a circle whose center is (4, -5) and which has a radius of 3.**

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.

Learn how to write the equation of a circle given its center (4, -5) and radius 3. This problem covers the standard form of a circle equation, including how to substitute the center and radius to find the equation. Suitable for students in Grades 9-12 studying geometry and circle equations.

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