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

Options

A. `0 lt c lt 1`
B. `c gt 2 `
C. ` c gt 0`
D. `c = pm sqrt3`

Answer

To solve the problem, we need to find the value of \( C \) given the direction cosines of a line as \( \left( \frac{1}{c}, \frac{1}{c}, \frac{1}{c} \right) \).

### Step-by-step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line are defined such that the sum of the squares of the direction cosines equals 1. This is a fundamental property of direction cosines.

2. **Setting Up the Equation**:
Given the direction cosines \( \left( \frac{1}{c}, \frac{1}{c}, \frac{1}{c} \right) \), we can set up the equation based on the property mentioned:
\[ \left( \frac{1}{c} \right)^2 + \left( \frac{1}{c} \right)^2 + \left( \frac{1}{c} \right)^2 = 1 \]

3. **Simplifying the Equation**:
This can be simplified as follows:
\[ 3 \left( \frac{1}{c^2} \right) = 1 \]

4. **Multiplying Both Sides by \( c^2 \)**:
To eliminate the fraction, multiply both sides by \( c^2 \):
\[ 3 = c^2 \]

5. **Taking the Square Root**:
Now, take the square root of both sides to solve for \( c \):
\[ c = \pm \sqrt{3} \]

6. **Conclusion**:
Therefore, the possible values of \( C \) are \( \sqrt{3} \) and \( -\sqrt{3} \).

### Final Answer:
The values of \( C \) are \( C = \sqrt{3} \) or \( C = -\sqrt{3} \).

---