If a variable plane in 3-dimensional space moves in such a way that the sum of the reciprocals of its intercepts on the x and y-axes exceeds the reciprocal of its intercept on the z-axis by 2, then all such planes will pass through the point:

Options

A. (1/2, 1/2, -1/2)
B. (1/2, 1/2, 1/2)
C. (1/2, -1/2, -1/2)
D. (1/2, -1/2, 1/2)

Answer

To solve the problem, we need to analyze the given condition about the variable plane in 3-dimensional space.

### Step-by-Step Solution:

1. **Understanding the Plane Equation**:
The equation of a plane that intercepts the x-axis at \( A \), the y-axis at \( B \), and the z-axis at \( C \) can be expressed as:
\[ \frac{x}{A} + \frac{y}{B} + \frac{z}{C} = 1 \]

2. **Reciprocals of Intercepts**:
The reciprocals of the intercepts are:
\[ \frac{1}{A}, \quad \frac{1}{B}, \quad \text{and} \quad \frac{1}{C} \]

3. **Setting Up the Condition**:
According to the problem, the sum of the reciprocals of the x and y intercepts exceeds the reciprocal of the z intercept by 2:
\[ \frac{1}{A} + \frac{1}{B} = \frac{1}{C} + 2 \]

4. **Rearranging the Equation**:
Rearranging the above equation gives:
\[ \frac{1}{A} + \frac{1}{B} - \frac{1}{C} = 2 \]

5. **Finding a Common Point**:
Let’s denote \( x_1 = 1 \), \( y_1 = 1 \), and \( z_1 = -1 \) as a point through which all such planes pass. We need to check if this point satisfies the condition derived:
\[ \frac{1}{A} + \frac{1}{B} - \frac{1}{C} = 2 \]

6. **Substituting the Point**:
Substitute \( x_1, y_1, z_1 \) into the plane equation:
\[ \frac{1}{A} + \frac{1}{B} + \frac{-1}{C} = 2 \]
This implies:
\[ \frac{1}{A} + \frac{1}{B} = 2 + \frac{1}{C} \]

7. **Finding the Ratios**:
From the condition, we can express the ratios:
\[ \frac{1}{A} = k, \quad \frac{1}{B} = k, \quad \frac{1}{C} = k - 2 \]
This gives us:
\[ A = \frac{1}{k}, \quad B = \frac{1}{k}, \quad C = \frac{1}{k - 2} \]

8. **Conclusion**:
Therefore, the point through which all such planes pass is:
\[ \left( \frac{1}{2}, \frac{1}{2}, -\frac{1}{2} \right) \]

### Final Answer:
The fixed point through which all such planes pass is:
\[ \boxed{\left( \frac{1}{2}, \frac{1}{2}, -\frac{1}{2} \right)} \]