What should be the value of k for which the equation `3x^2+3y^2+(k+1)z^2+x-y+z=0` represents the sphere

Options

A. 3
B. 2
C. 1
D. `-1`

Answer

To determine the value of \( k \) for which the equation

\[ 3x^2 + 3y^2 + (k+1)z^2 + x - y + z = 0 \]

represents a sphere, we need to ensure that the coefficients of \( x^2 \), \( y^2 \), and \( z^2 \) are equal.

### Step-by-Step Solution:

1. **Identify the coefficients of \( x^2 \), \( y^2 \), and \( z^2 \)**:
- The coefficient of \( x^2 \) is \( 3 \).
- The coefficient of \( y^2 \) is \( 3 \).
- The coefficient of \( z^2 \) is \( k + 1 \).

2. **Set the coefficients equal to each other**:
- For the equation to represent a sphere, the coefficients must be equal:
\[ 3 = 3 = k + 1 \]

3. **Solve for \( k \)**:
- From the equation \( k + 1 = 3 \), we can isolate \( k \):
\[ k + 1 = 3 \implies k = 3 - 1 \]
\[ k = 2 \]

4. **Conclusion**:
- The value of \( k \) that makes the equation represent a sphere is \( k = 2 \).