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.

Would you like more details or have any questions?

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.