explain "Since the x-axis is a common tangent and given the radii, the distances from the centers to the x-axis must be equal to the radii:"

Sure, let's delve into why "Since the x-axis is a common tangent and given the radii, the distances from the centers to the x-axis must be equal to the radii."

### Explanation:

1. **Definition of a Tangent Line:**
   - A tangent to a circle is a line that touches the circle at exactly one point.
   - At the point of tangency, the tangent line is perpendicular to the radius of the circle.

2. **X-Axis as a Tangent:**
   - If the x-axis is a tangent to the circle, it touches the circle at exactly one point along the x-axis (where \( y = 0 \)).
   - The point of tangency is where the circle just touches the x-axis without crossing it.

3. **Relationship Between Radius and Tangent Line:**
   - The radius of the circle at the point of tangency is perpendicular to the tangent line.
   - Therefore, the distance from the center of the circle to the x-axis (which is the radius) must be exactly equal to the radius of the circle.
   
4. **Centers and Radii:**
   - Let’s consider the circle with center \((a, b)\) and radius \(r\).
   - Since the x-axis is a tangent, the vertical distance from the center \((a, b)\) to the x-axis (where \( y = 0 \)) must be equal to the radius \(r\).
   - This vertical distance is simply the y-coordinate of the center, \(b\).

5. **Applying to Given Radii:**
   - For circle 1, given the radius \( r_1 = 5 \):
     - The distance from the center of the circle to the x-axis must be 5.
     - Therefore, the y-coordinate of the center, \( b_1 = 5 \).
   - For circle 2, given the radius \( r_2 = 10 \):
     - The distance from the center of the circle to the x-axis must be 10.
     - Therefore, the y-coordinate of the center, \( b_2 = 10 \).

### Summary

Since the x-axis is a tangent to the circles, the y-coordinate of the centers of the circles must be equal to the radii of the circles. This ensures that the distance from each center to the x-axis is exactly equal to the radius, maintaining the tangency condition.

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### Relative Questions

1. How do you find the center of a circle given its equation?
2. What is the general form of the equation of a circle?
3. How do you determine if a line is tangent to a circle?
4. Can two circles have more than one common tangent? Explain.
5. How do you derive the distance formula between a point and a line?
6. What are the properties of circles with a common tangent?
7. How do you find the points of tangency of a tangent line to a circle?
8. What are some applications of tangent lines in real-world problems?

### Tip

When solving geometric problems involving circles, always draw a diagram. It helps visualize relationships and understand the problem better.

Explore the concept of tangent lines to circles in geometry. Learn why the distances from the centers to the x-axis must equal the radii when the x-axis is a common tangent. Suitable for high school students studying circle geometry.

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