find the equation of sphere passes through the points (4,1,0),(2,-3,4) and (1,0,0) and touch the plane 2x+2y-z=11.

To find the equation of a sphere passing through the points \((4,1,0)\), \((2,-3,4)\), \((1,0,0)\), and touching the plane \(2x + 2y - z = 11\), follow these steps:

### General Equation of a Sphere

The general equation of a sphere is:

\[
(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2
\]

Where:
- \((x_0, y_0, z_0)\) is the center of the sphere,
- \(r\) is the radius.

Alternatively, it can be expanded as:

\[
x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0
\]

Where \(D\), \(E\), \(F\), and \(G\) are constants to be determined.

### Step 1: Apply the Points on the Sphere

Since the sphere passes through the points \((4,1,0)\), \((2,-3,4)\), and \((1,0,0)\), they must satisfy the equation of the sphere. Substituting these points into the general equation \(x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0\):

1. For \((4,1,0)\):
   \[
   4^2 + 1^2 + 0^2 + 4D + 1E + 0F + G = 0
   \]
   \[
   17 + 4D + E + G = 0 \quad \text{(Equation 1)}
   \]

2. For \((2,-3,4)\):
   \[
   2^2 + (-3)^2 + 4^2 + 2D + (-3)E + 4F + G = 0
   \]
   \[
   29 + 2D - 3E + 4F + G = 0 \quad \text{(Equation 2)}
   \]

3. For \((1,0,0)\):
   \[
   1^2 + 0^2 + 0^2 + 1D + 0E + 0F + G = 0
   \]
   \[
   1 + D + G = 0 \quad \text{(Equation 3)}
   \]

### Step 2: Apply the Condition for Tangency

The sphere touches the plane \(2x + 2y - z = 11\). The distance from the center of the sphere \((x_0, y_0, z_0)\) to the plane \(Ax + By + Cz + D = 0\) is given by:

\[
\frac{|Ax_0 + By_0 + Cz_0 - D|}{\sqrt{A^2 + B^2 + C^2}} = r
\]

For the plane \(2x + 2y - z = 11\), we have:
- \(A = 2\),
- \(B = 2\),
- \(C = -1\),
- Constant term \(D = -11\).

This gives us another equation relating the center \((x_0, y_0, z_0)\) and the radius \(r\).

### Step 3: Solve the System of Equations

From these four equations, solve for \(D\), \(E\), \(F\), and \(G\), and find the equation of the sphere.

I'll solve this system step by step.

The solution to the system of equations gives the following relationships for \(D\), \(E\), and \(F\) in terms of \(G\):

- \(D = -G - 1\),
- \(E = 3G - 13\),
- \(F = \frac{5G}{2} - \frac{33}{2}\).

Now, we need to apply the tangency condition to determine the value of \(G\), and subsequently \(D\), \(E\), and \(F\). Using the distance formula from the center of the sphere to the plane \(2x + 2y - z = 11\), I'll solve for \(G\) next.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to walk you through the remaining steps to find the value of \(G\), or answer any other questions about the process?

Here are 5 related questions to expand on this topic:
1. How do we determine the center of a sphere given different conditions?
2. What is the geometric meaning of a sphere touching a plane?
3. How can we generalize the equation of a sphere in 3D space?
4. How is the distance from a point to a plane calculated in 3D?
5. How do constraints, like tangency, affect the parameters of geometric objects?

**Tip:** When solving systems of equations with geometry, keep the relationships between different elements (like points, planes, and distances) clear to avoid mistakes.

Learn how to find the equation of a sphere that passes through the points (4,1,0), (2,-3,4), and (1,0,0) while touching the plane 2x + 2y - z = 11. This problem involves key concepts in spherical geometry and the application of distance formulas in coordinate geometry. Suitable for college-level students.

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