If direction cosines of a line are `(1/a, 1/a, 1/a)` then

Options

A. `agt0`
B. `0lt alt 1`
C. `a = pm sqrt(3)`
D. `a gt 2`

Answer

To solve the problem, we need to analyze the given direction cosines of a line, which are \( \left( \frac{1}{a}, \frac{1}{a}, \frac{1}{a} \right) \).

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line are defined as \( L = \cos \alpha \), \( M = \cos \beta \), and \( N = \cos \gamma \), where \( \alpha \), \( \beta \), and \( \gamma \) are the angles that the line makes with the coordinate axes.

2. **Using the Property of Direction Cosines**:
The fundamental property of direction cosines is that:
\[ L^2 + M^2 + N^2 = 1 \]
In our case, we have:
\[ L = \frac{1}{a}, \quad M = \frac{1}{a}, \quad N = \frac{1}{a} \]

3. **Substituting the Values**:
Substitute the values of \( L \), \( M \), and \( N \) into the equation:
\[ \left( \frac{1}{a} \right)^2 + \left( \frac{1}{a} \right)^2 + \left( \frac{1}{a} \right)^2 = 1 \]

4. **Simplifying the Equation**:
This simplifies to:
\[ \frac{1}{a^2} + \frac{1}{a^2} + \frac{1}{a^2} = 1 \]
Which can be written as:
\[ 3 \cdot \frac{1}{a^2} = 1 \]

5. **Solving for \( a^2 \)**:
To isolate \( a^2 \), we multiply both sides by \( a^2 \):
\[ 3 = a^2 \]
Thus, we find:
\[ a^2 = 3 \]

6. **Finding \( a \)**:
Taking the square root of both sides gives:
\[ a = \pm \sqrt{3} \]

### Final Answer:
The value of \( a \) is \( \pm \sqrt{3} \).

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