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

Explanation

To determine the condition 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 that the sum of the squares of the direction cosines must equal 1.

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line are represented as \(l, m, n\). For a line with direction cosines \(l, m, n\), the condition is:
\[ l^2 + m^2 + n^2 = 1 \]
In our case, we have:
\[ l = \frac{1}{\sqrt{2}}, \quad m = \frac{1}{2}, \quad n = K \]

2. **Setting Up the Equation**:
Substitute the values of \(l\), \(m\), and \(n\) into the equation:
\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + K^2 = 1 \]

3. **Calculating the Squares**:
Calculate each term:
- \(\left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}\)
- \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)

So, the equation becomes:
\[ \frac{1}{2} + \frac{1}{4} + K^2 = 1 \]

4. **Finding a Common Denominator**:
To combine the fractions, find a common denominator (which is 4):
\[ \frac{2}{4} + \frac{1}{4} + K^2 = 1 \]

5. **Simplifying the Equation**:
Combine the fractions:
\[ \frac{3}{4} + K^2 = 1 \]

6. **Isolating \(K^2\)**:
Subtract \(\frac{3}{4}\) from both sides:
\[ K^2 = 1 - \frac{3}{4} \]
\[ K^2 = \frac{1}{4} \]

7. **Finding \(K\)**:
Taking the square root of both sides gives:
\[ K = \pm \frac{1}{2} \]

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