The equation of the circle can be derived from the graph. The key components to identify are:

1. Center: The center of the circle appears to be at (5, -4) based on the location in the coordinate plane.
2. Radius: The radius is the distance from the center (5, -4) to any point on the circle.

Using the standard equation of a circle:

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

Where:

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

Step-by-step process:

1. Center: From the graph, the center is $(5, -4)$, so $h = 5$ and $k = -4$.

2. Radius: The radius can be determined by measuring the distance from the center to a point on the circle. From the graph, the radius is approximately $\sqrt{34}$.

3. Equation: Substituting into the standard form: $(x - 5)^2 + (y + 4)^2 = 34$

Answer: The equation of the circle matches Option A: $(x - 5)^2 + (y + 4)^2 = 34$.

If you have further questions or would like to explore the properties of this circle, feel free to ask!

5 Related Questions:

1. How can you calculate the radius of a circle given its center and a point on the circle?
2. What are the key differences between standard and general forms of a circle's equation?
3. How does shifting the center of a circle affect its equation?
4. What is the geometric meaning of completing the square in a circle equation?
5. How would you determine if a given point lies inside, outside, or on the circle?

Tip:

To verify the equation of a circle, substitute the center coordinates and a point on the circle into the equation to ensure they satisfy it.