Find equations of the spheres with center (2,-3,6) that touch the xy-plane

To find the equations of the spheres that have their center at \((2, -3, 6)\) and touch the \(xy\)-plane, we follow these steps:

### 1. **Understanding the Problem:**
   - The center of the sphere is given by the point \((2, -3, 6)\).
   - The sphere touches the \(xy\)-plane, meaning the distance from the center of the sphere to the \(xy\)-plane is equal to the radius \(r\) of the sphere.

### 2. **Distance from the Center to the \(xy\)-Plane:**
   - The \(xy\)-plane is characterized by the equation \(z = 0\).
   - The distance from the point \((2, -3, 6)\) to the \(xy\)-plane is simply the absolute value of the \(z\)-coordinate of the center.
     \[
     \text{Distance} = |6| = 6
     \]
   - Therefore, the radius \(r\) of the sphere is \(6\).

### 3. **Equation of the Sphere:**
   - The general equation of a sphere with center \((h, k, l)\) and radius \(r\) is given by:
     \[
     (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
     \]
   - Substituting \(h = 2\), \(k = -3\), \(l = 6\), and \(r = 6\), the equation becomes:
     \[
     (x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 6^2
     \]
     \[
     (x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 36
     \]

### **Conclusion:**
The equation of the sphere is:
\[
(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 36
\]

This sphere has its center at \((2, -3, 6)\) and touches the \(xy\)-plane.

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Here are some related questions to explore:
1. What is the equation of a sphere centered at \((h, k, l)\) with a different radius \(r\)?
2. How would the equation change if the sphere touched the \(yz\)-plane instead?
3. How can you determine if a given point lies on this sphere?
4. What is the distance between two points in 3-dimensional space?
5. How can we find the equation of a sphere that passes through a specific point?

**Tip:** Always remember that the radius of a sphere touching a plane is the perpendicular distance from the center to that plane.

Explore how to find the equations of spheres with centers at (2, -3, 6) that touch the xy-plane. Includes step-by-step instructions and formulas for determining the sphere's radius and center. Suitable for high school students.

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