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 use the property of direction cosines. The direction cosines of a line are given as \(\left(\frac{1}{c}, \frac{1}{c}, \frac{1}{c}\right)\). We know that the sum of the squares of the direction cosines must equal 1.

### Step-by-step Solution:

1. **Write the equation for the sum of squares of direction cosines**:
\[ \left(\frac{1}{c}\right)^2 + \left(\frac{1}{c}\right)^2 + \left(\frac{1}{c}\right)^2 = 1 \]

2. **Simplify the left-hand side**:
\[ 3 \left(\frac{1}{c}\right)^2 = 1 \]

3. **Multiply both sides by \(c^2\)** to eliminate the fraction:
\[ 3 = c^2 \]

4. **Solve for \(c\)** by taking the square root of both sides:
\[ c = \pm \sqrt{3} \]

5. **Conclusion**:
The values of \(c\) are \(c = \sqrt{3}\) and \(c = -\sqrt{3}\).

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

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