This is a question about a circle given by the equation:

$(x - 2)^2 + (y + 9)^2 = 64$

Key details of the equation:

1. Standard form of a circle: $(x - h)^2 + (y - k)^2 = r^2$

   • The center of the circle is $(h, k)$.
   • The radius of the circle is $r$, obtained by taking the square root of $r^2$.

2. Given equation:

   • $(x - 2)^2 + (y + 9)^2 = 64$
   • Here, $h = 2$, $k = -9$, and $r^2 = 64$, so $r = \sqrt{64} = 8$.

Characteristics of the circle:

• Center: $(2, -9)$
• Radius: $8$

To identify the correct graph:

• Look for a circle centered at $(2, -9)$ with a radius extending $8$ units in all directions.

Would you like me to elaborate on how this would look visually on a graph or provide more examples?

Related Questions:

1. How do you determine the center of a circle from its equation?
2. What changes in the graph when the radius of a circle is doubled?
3. What does the term $(x - h)^2 + (y - k)^2$ geometrically represent?
4. How does the position of the center shift if $k$ becomes positive?
5. Can this equation be rewritten in another form, such as parametric form?

Tip: Always rewrite the circle equation in its standard form to quickly identify the center and radius.