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.