Question

If the line \(x - 1 = 0\) is the directrix of the parabola \(y^2 - kx + 8 = 0\), then one of the value of \(k\) is

• A. \(\frac{1}{8}\)
• B. 8
• C. 4
• D. 1

Hint:

For parabola \(y^2 = 4a(x - h)\), vertex at \((h,0)\), directrix \(x = h - a\).

Answer 1

The Correct Option is C

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

A parabola with horizontal axis can be represented by the general equation:

\(y^2 = 4a(x - h)\)

Here, the vertex is at \((h, 0)\), the focus is at \((h+a, 0)\), and the directrix is the line \(x = h-a\).

Comparing this to the given equation of the parabola, \(y^2 - kx + 8 = 0\), we can rewrite this as:

\(y^2 = kx - 8\) or \(y^2 = k\left(x - \frac{8}{k}\right)\)

This matches the form \(y^2 = 4a(x - h)\) where:

• \(4a = k\)
• \(h = \frac{8}{k}\)

Since the given directrix is \(x = 1\), we equate it to the directrix \(x = h - a\):

\(h - a = 1\)

Substituting the values of \(h\) and \(a\) in terms of \(k\):

\(\frac{8}{k} - \frac{k}{4} = 1\)

Solving this equation will give us the value of \(k\):

1. Substitute \(h = \frac{8}{k}\) and \(a = \frac{k}{4}\) into the equation:
2. \(\frac{8}{k} - \frac{k}{4} = 1\)
3. Multiply the entire equation by \(4k\) to eliminate the denominators:
4. \(4 \times 8 - k^2 = 4k\)
5. Simplify:
6. \(32 - k^2 = 4k\)
7. Reorder the equation:
8. \(k^2 + 4k - 32 = 0\)

This is a quadratic equation in the form \(k^2 + 4k - 32 = 0\).

We use the quadratic formula, where \(a = 1\), \(b = 4\), and \(c = -32\):

\(k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

\(k = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-32)}}{2 \cdot 1}\)

\(k = \frac{-4 \pm \sqrt{16 + 128}}{2}\)

\(k = \frac{-4 \pm \sqrt{144}}{2}\)

\(k = \frac{-4 \pm 12}{2}\)

1. Calculate the two possible values for \(k\):
2. \(k_1 = \frac{-4 + 12}{2} = 4\)
3. \(k_2 = \frac{-4 - 12}{2} = -8\)

Among the given options, one of the values of \(k\) that matches is \(k = 4\).

Therefore, the correct answer is 4.