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Part 1

The equation of a circle is given in standard form. Complete parts a through c to follow the step by step process to sketch the graph of the circle.

left parenthesis x plus 5 right parenthesis squared plus left parenthesis y plus 6 right parenthesis squared equals 81(x+5)2+(y+6)2=81

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Part 1

​a)**** Determine the center and radius of the circle.

The center of the circle is 

enter your response here.

The standard form of the equation of a circle is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Where:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius.

Given the equation:

\[
(x + 5)^2 + (y + 6)^2 = 81
\]

This can be rewritten as:

\[
(x - (-5))^2 + (y - (-6))^2 = 81
\]

By comparing this with the standard form, we can determine:
- The center is \((-5, -6)\).
- The radius is \(\sqrt{81} = 9\).

### Final Answer:
- The center of the circle is \((-5, -6)\).
- The radius is \(9\).

Would you like further explanation or have any other questions? Here are some related questions to consider:

1. What is the general method for finding the center of a circle from its equation?
2. How do you sketch the graph of a circle given its center and radius?
3. Can you derive the equation of a circle from given points on the circumference?
4. What changes in the equation if the circle is shifted to a new location?
5. How do you find the equation of a circle given its diameter?

**Tip:** Always ensure the equation is in standard form before identifying the center and radius.

The equation of a circle is given in standard form. Complete parts a through c to follow the step-by-step process to sketch the graph of the circle.

(x + 5)^2 + (y + 6)^2 = 81

Part a) Determine the center and radius of the circle.

Learn how to determine the center and radius of a circle given its equation in standard form. This step-by-step solution explains key concepts in coordinate geometry and circle equations. Suitable for students in Grades 9-12.

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