The intersection of the spheres `x^2+y^2+z^2+7x-2y-z=13a n dx^2+y^2+z^2-3x+3y+4z=8` is the same as the intersection of one of the spheres and the plane a. `x-y-z=1` b. `x-2y-z=1` c. `x-y-2z=1` d. `2x-y-z=1`

Options

A. `x-y-z=1`
B. `x-2y-z=1`
C. `x-y-2z=1`
D. `2x-y-z=1`

Explanation

To solve the problem, we need to find the intersection of the two given spheres and determine which of the provided planes represents the same intersection.

### Step 1: Write down the equations of the spheres
The equations of the spheres are given as:
1. \( S: x^2 + y^2 + z^2 + 7x - 2y - z = 13 \)
2. \( S': x^2 + y^2 + z^2 - 3x + 3y + 4z = 8 \)

### Step 2: Subtract the equations of the spheres
To find the intersection of the spheres, we can subtract one equation from the other:
\[ S - S' = 0 \]
This gives us:
\[ (x^2 + y^2 + z^2 + 7x - 2y - z - 13) - (x^2 + y^2 + z^2 - 3x + 3y + 4z - 8) = 0 \]

### Step 3: Simplify the equation
Now, simplifying the above expression:
\[ 7x - 2y - z - 13 + 3x - 3y - 4z + 8 = 0 \]
Combining like terms:
\[ (7x + 3x) + (-2y - 3y) + (-z - 4z) + (-13 + 8) = 0 \]
This simplifies to:
\[ 10x - 5y - 5z - 5 = 0 \]

### Step 4: Factor out common terms
We can factor out 5 from the equation:
\[ 5(2x - y - z - 1) = 0 \]
Thus, we have:
\[ 2x - y - z = 1 \]

### Step 5: Identify the corresponding plane
Now, we need to check which of the given options matches this equation:
- a. \( x - y - z = 1 \)
- b. \( x - 2y - z = 1 \)
- c. \( x - y - 2z = 1 \)
- d. \( 2x - y - z = 1 \)

From our derived equation, we see that option **d** matches perfectly:
\[ 2x - y - z = 1 \]

### Conclusion
The correct answer is option **d**: \( 2x - y - z = 1 \).