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The question asks for the standard form of the equation of a circle with a center at \( (3, -2) \) and radius 4.

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.

In this case, the center is \( (3, -2) \), so \( h = 3 \) and \( k = -2 \), and the radius is \( 4 \), so \( r = 4 \).

Now substitute these values into the formula:

\[
(x - 3)^2 + (y - (-2))^2 = 4^2
\]

Simplifying:

\[
(x - 3)^2 + (y + 2)^2 = 16
\]

So, the correct equation is:

\[
(x - 3)^2 + (y + 2)^2 = 16
\]

The correct answer is the second option:
\[
\boxed{(x - 3)^2 + (y + 2)^2 = 16}
\]

Would you like more details on how this equation is derived, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. How do you determine the radius of a circle from its equation?
2. What is the general form of a circle's equation, and how does it relate to the standard form?
3. How would the equation change if the radius were different, say 5 instead of 4?
4. Can you derive the equation of a circle from its graph?
5. What would the equation of a circle be if its center were shifted to \( (-3, 2) \)?

**Tip:** When writing the equation of a circle, always remember to carefully check the signs for the center coordinates in the formula \( (x - h)^2 + (y - k)^2 = r^2 \).

What is the standard form of the equation of a circle with center (3, -2) and radius 4?

Find the standard form of a circle equation with center at (3, -2) and radius 4. This question demonstrates key geometry concepts in circle equations for students in Grades 9-10.

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