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

To find the equations of the spheres that have their center at \((2, -3, 6)\) and touch the \(yz\)-plane, let's proceed with the following steps:

### 1. **Understanding the Problem:**
   - The center of the sphere is given as \((2, -3, 6)\).
   - The sphere touches the \(yz\)-plane, which means the distance from the center of the sphere to the \(yz\)-plane is equal to the radius \(r\) of the sphere.

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

### 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 \(r = 2\), the equation becomes:
     \[
     (x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 2^2
     \]
     \[
     (x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 4
     \]

### **Conclusion:**
The equation of the sphere is:
\[
(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 4
\]

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

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Would you like more details on any part of this solution? 

Here are some related questions to explore:
1. What changes in the equation if the sphere touches the \(xz\)-plane instead?
2. How can we derive the equation of a sphere given two points on its diameter?
3. What is the geometric interpretation of a sphere that touches more than one coordinate plane?
4. How do you find the point of contact between a sphere and a plane?
5. What is the equation of a sphere if it passes through a specific point and touches the \(yz\)-plane?

**Tip:** When a sphere touches a coordinate plane, the radius is always equal to the absolute value of the coordinate corresponding to that axis.

Discover how to find the equations of spheres that touch the yz-plane with a center at (2, -3, 6). This problem explores the geometry of spheres in coordinate space, covering the equation derivation and application. Suitable for advanced high school students and beyond.

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