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The image shows a math problem related to the equation of a circle. Here is the content extracted from the image:

1. **Determine the coordinates of the center and radius of the circle from the equation given below:**
   \[ x^2 + y^2 = 72 \]

2. **Find the standard equation of the circle which satisfies the given conditions:**
   - Center at \( (3, 7) \)
   - Tangent to the x-axis
   - Another condition: \( x^2 + y^2 = 72 \)

### Solution to Part 1:

The general equation of a circle in standard form is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center of the circle and \(r\) is the radius.

The given equation is:
\[
x^2 + y^2 = 72
\]

This equation can be rewritten in the standard form as:
\[
(x - 0)^2 + (y - 0)^2 = 72
\]

From this, we see that:
- The center of the circle is at \((0, 0)\).
- The radius \(r\) is \(\sqrt{72} = 6\sqrt{2}\).

### Solution to Part 2:

Now, for the second part:

- The center of the circle is given as \( (3, 7) \).
- The circle is tangent to the x-axis.

For the circle to be tangent to the x-axis, the distance from the center \((3, 7)\) to the x-axis must be equal to the radius \(r\). The distance from a point \((x_1, y_1)\) to the x-axis is simply the absolute value of its y-coordinate, \(|y_1|\).

Here, the y-coordinate of the center is 7, so the radius \(r = 7\).

Now, the equation of the circle in standard form is:
\[
(x - 3)^2 + (y - 7)^2 = 7^2
\]

Simplifying, we get:
\[
(x - 3)^2 + (y - 7)^2 = 49
\]

### Summary:
1. **Coordinates of the center and radius of the first circle:**
   - Center: \((0, 0)\)
   - Radius: \(6\sqrt{2}\)

2. **Equation of the second circle:**
   - Center: \((3, 7)\)
   - Radius: \(7\)
   - Equation: \((x - 3)^2 + (y - 7)^2 = 49\)

Would you like more details or have any questions?

### Related Questions:
1. How do you find the equation of a circle tangent to both axes?
2. How is the distance from a point to a line (such as the x-axis) calculated?
3. What is the geometric interpretation of a circle being tangent to a line?
4. Can there be more than one circle that satisfies a given set of conditions?
5. How do we determine if two circles are tangent to each other?

### Tip:
Remember, a circle tangent to the x-axis has its radius equal to the absolute value of its y-coordinate.

Explore how to solve the equation x^2 + y^2 = 72 with a step-by-step graphical illustration. Understand the concept of the circle's equation, its center, and radius. Suitable for high school students studying coordinate geometry.

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