A plane passing through `(1,1,1)` cuts positive direction of coordinates axes at `A ,Ba n dC ,` then the volume of tetrahedron `O A B C` satisfies a. `Vlt=9/2` b. `Vgeq9/2` c. `V=9/2` d. none of these

Options

A. `Vle(9)/(2)`
B. `Vge(9)/(2)`
C. `V=(9)/(2)`
D. none of these

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Geometry of the Problem
We have a plane that passes through the point \( (1, 1, 1) \) and intersects the positive axes at points \( A \), \( B \), and \( C \). The coordinates of these points can be represented as:
- \( A(a, 0, 0) \)
- \( B(0, b, 0) \)
- \( C(0, 0, c) \)

### Step 2: Equation of the Plane
The equation of a plane that intersects the axes at points \( A \), \( B \), and \( C \) can be expressed in the form:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
Since the plane passes through the point \( (1, 1, 1) \), we can substitute these coordinates into the equation:
\[ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1 \]

### Step 3: Volume of the Tetrahedron
The volume \( V \) of the tetrahedron \( OABC \) is given by the formula:
\[ V = \frac{1}{6} \times a \times b \times c \]

### Step 4: Applying the AM-GM Inequality
To find a relationship between \( a \), \( b \), and \( c \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
From the equation \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1 \), we can rearrange it to express \( a, b, c \):
\[ \frac{bc + ac + ab}{abc} = 1 \implies bc + ac + ab = abc \]

### Step 5: Using AM-GM on \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \)
Using the AM-GM inequality on \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \):
\[ \frac{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}{3} \geq \sqrt[3]{\frac{1}{abc}} \]
Since \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1 \), we have:
\[ \frac{1}{3} \geq \sqrt[3]{\frac{1}{abc}} \implies abc \geq 27 \]

### Step 6: Calculate the Volume
Substituting \( abc \geq 27 \) into the volume formula:
\[ V = \frac{1}{6} \times abc \geq \frac{1}{6} \times 27 = \frac{27}{6} = \frac{9}{2} \]

### Conclusion
Thus, the volume of the tetrahedron \( OABC \) satisfies:
\[ V \geq \frac{9}{2} \]
The correct answer is option **b**: \( V \geq \frac{9}{2} \).

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