If the direction cosines of a line are `lt k, k, kgt` then

Options

A. `k gt 0`
B. ` 0 lt k lt 1`
C. `k=1`
D. `k=(1)/(sqrt(3) )` or `k=- (1)/( sqrt3)`

Answer

To solve the problem, we need to analyze the given direction cosines of the line, which are \( k, k, k \). We will use the property that the sum of the squares of the direction cosines of a line is equal to 1.

### Step-by-Step Solution:

1. **Identify the direction cosines**: The direction cosines of the line are given as \( l = k \), \( m = k \), and \( n = k \).

2. **Apply the property of direction cosines**: According to the property, the sum of the squares of the direction cosines must equal 1:
\[ l^2 + m^2 + n^2 = 1 \]

3. **Substitute the values**: Substitute \( l, m, n \) with \( k \):
\[ k^2 + k^2 + k^2 = 1 \]

4. **Combine like terms**: This simplifies to:
\[ 3k^2 = 1 \]

5. **Solve for \( k^2 \)**: Divide both sides by 3:
\[ k^2 = \frac{1}{3} \]

6. **Take the square root**: To find \( k \), take the square root of both sides:
\[ k = \pm \frac{1}{\sqrt{3}} \]

### Conclusion:
The possible values for \( k \) are \( k = \frac{1}{\sqrt{3}} \) or \( k = -\frac{1}{\sqrt{3}} \). Therefore, the correct option is the fourth one.