If the line `x-1=0` is the directrix of the parabola `y^2-k x+8=0` , then one of the values of `k` is `1/8` (b) 8 (c) 4 (d) `1/4`

Options

A. -8
B. `1//8`
C. `1//4`
D. 4

Explanation

To solve the problem, we need to find the value of \( k \) such that the line \( x - 1 = 0 \) (or \( x = 1 \)) is the directrix of the parabola given by the equation \( y^2 - kx + 8 = 0 \).

### Step-by-Step Solution:

1. **Rearranging the Parabola Equation:**
We start with the equation of the parabola:
\[ y^2 - kx + 8 = 0 \]
Rearranging gives:
\[ y^2 = kx - 8 \]
This can be rewritten as:
\[ y^2 = k\left(x - \frac{8}{k}\right) \]
This shows that the vertex of the parabola is at \( \left(\frac{8}{k}, 0\right) \).

**Hint:** Identify the vertex from the standard form of the parabola.

2. **Finding the Distance from the Vertex to the Directrix:**
The distance \( a \) from the vertex to the directrix can be expressed as:
\[ a = \frac{k}{4} \]
The directrix is the line \( x = 1 \), so the distance from the vertex \( \left(\frac{8}{k}, 0\right) \) to the line \( x = 1 \) is given by:
\[ \left|\frac{8}{k} - 1\right| \]

**Hint:** Use the formula for the distance from a point to a line.

3. **Setting Up the Equation:**
Since the distance from the vertex to the directrix must equal \( a \), we have:
\[ \frac{k}{4} = \left|\frac{8}{k} - 1\right| \]
We can remove the absolute value since we know the vertex is to the right of the directrix (for positive \( k \)):
\[ \frac{k}{4} = \frac{8}{k} - 1 \]

**Hint:** Consider the properties of the parabola to determine the sign of the distance.

4. **Solving the Equation:**
Multiply both sides by \( 4k \) to eliminate the fraction:
\[ k^2 = 32 - 4k \]
Rearranging gives:
\[ k^2 + 4k - 32 = 0 \]

**Hint:** Rearranging the equation helps in standard quadratic form.

5. **Factoring the Quadratic:**
We can factor the quadratic:
\[ (k + 8)(k - 4) = 0 \]
This gives us two possible solutions for \( k \):
\[ k + 8 = 0 \quad \Rightarrow \quad k = -8 \quad \text{(not valid since } k \text{ must be positive)} \]
\[ k - 4 = 0 \quad \Rightarrow \quad k = 4 \]

**Hint:** Factorization can simplify finding the roots of a quadratic equation.

6. **Conclusion:**
The valid value of \( k \) is:
\[ k = 4 \]

Therefore, one of the values of \( k \) is \( 4 \).

### Final Answer:
The correct option is (c) \( 4 \).