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The question in the image asks for the equation of a circle in the xy-plane that has its center at \((0, 0)\) and a radius \(r = 4\).

### Solution:

The general equation of a circle with center \((h, k)\) and radius \(r\) is:

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

Given that the center is at \((0, 0)\), we substitute \(h = 0\) and \(k = 0\), giving us:

\[
x^2 + y^2 = r^2
\]

Since the radius \(r = 4\), we substitute \(r\) with 4:

\[
x^2 + y^2 = 4^2
\]

This simplifies to:

\[
x^2 + y^2 = 16
\]

### Answer:
The equation of the circle with center \((0, 0)\) and radius \(4\) is:

\[
x^2 + y^2 = 16
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you derive the equation of a circle given a different center and radius?
2. What is the distance formula, and how does it relate to the equation of a circle?
3. How do you find the points of intersection between two circles?
4. What are the parametric equations for a circle centered at the origin?
5. How does changing the center or radius of a circle affect its equation?

### Tip:
Remember, the equation of a circle in standard form provides direct information about its center and radius. The squared terms for \(x\) and \(y\) indicate a circle, with the center and radius clearly defined by the coefficients.

Which of the following is an equation of a circle in the xy-plane that has center (0, 0) and radius r = 4?

This problem helps you find the equation of a circle with center at (0,0) and radius 4. Using the standard equation of a circle, you will learn how to apply algebraic substitution to simplify and solve circle equations. Suitable for high school students in Grades 9-12.

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