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

Options

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

Explanation

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

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

represents a sphere, we need to analyze the conditions under which a quadratic equation in three variables represents a sphere.

### Step-by-Step Solution:

1. **Identify the General Form of a Sphere**:
The general equation of a sphere in 3D is given by the form:

\[ Ax^2 + By^2 + Cz^2 + Dx + Ey + F = 0 \]

For this to represent a sphere, the coefficients of \( x^2 \), \( y^2 \), and \( z^2 \) must be equal, i.e., \( A = B = C \).

2. **Extract Coefficients from the Given Equation**:
From the equation \( 3x^2 + 2y^2 + (k+1)z^2 + x - y + z = 0 \), we can identify the coefficients:
- Coefficient of \( x^2 \) (A) = 3
- Coefficient of \( y^2 \) (B) = 2
- Coefficient of \( z^2 \) (C) = \( k + 1 \)

3. **Set Up the Equality Condition**:
For the equation to represent a sphere, we need:

\[ A = B = C \]

This gives us two equations:
- \( A = B \) implies \( 3 = 2 \) (which is not true, so we need to set \( C \) to 3)
- \( A = C \) implies \( 3 = k + 1 \)

4. **Solve for \( k \)**:
From the equation \( 3 = k + 1 \), we can solve for \( k \):

\[ k + 1 = 3 \implies k = 3 - 1 \implies k = 2 \]

5. **Conclusion**:
Therefore, the value of \( k \) for which the equation represents a sphere is

\[ k = 2. \]