Under what condition do `<<1/(sqrt2).""1/2.K>>` represent direction cosines of a line ?

Options

A. `k=1/2`
B. `k=-1/2`
C. `k=pm1/2`
D. k can take any value

Answer

To determine the conditions under which the values \( \frac{1}{\sqrt{2}}, \frac{1}{2}, k \) represent the direction cosines of a line, we need to use the property of direction cosines. The direction cosines \( l, m, n \) must satisfy the equation:

\[ l^2 + m^2 + n^2 = 1 \]

### Step-by-Step Solution:

1. **Assign Values**:
- Let \( l = \frac{1}{\sqrt{2}} \)
- Let \( m = \frac{1}{2} \)
- Let \( n = k \)

2. **Substitute into the Condition**:
- Substitute the values into the equation:
\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + k^2 = 1 \]

3. **Calculate Each Term**:
- Calculate \( l^2 \):
\[ \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]
- Calculate \( m^2 \):
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

4. **Combine the Terms**:
- Substitute these values back into the equation:
\[ \frac{1}{2} + \frac{1}{4} + k^2 = 1 \]

5. **Find a Common Denominator**:
- The least common multiple (LCM) of 2 and 4 is 4. Rewrite \( \frac{1}{2} \) as \( \frac{2}{4} \):
\[ \frac{2}{4} + \frac{1}{4} + k^2 = 1 \]

6. **Combine the Fractions**:
- Combine the fractions:
\[ \frac{2 + 1}{4} + k^2 = 1 \]
\[ \frac{3}{4} + k^2 = 1 \]

7. **Isolate \( k^2 \)**:
- Subtract \( \frac{3}{4} \) from both sides:
\[ k^2 = 1 - \frac{3}{4} \]
\[ k^2 = \frac{1}{4} \]

8. **Solve for \( k \)**:
- Take the square root of both sides:
\[ k = \pm \frac{1}{2} \]

### Conclusion:
The values \( \frac{1}{\sqrt{2}}, \frac{1}{2}, k \) represent the direction cosines of a line when \( k = \frac{1}{2} \) or \( k = -\frac{1}{2} \).