If the line is the directrix of the parabola , then one of the values of is
- A.
- B.
- C.
- D.
Show answer and solution
Correct option: C
Solution
Rewrite the parabola: y² = kx - 8 = k(x - 8/k), so vertex is at (8/k, 0) and the equation in standard form is y² = k·(x - 8/k).
For y² = 4a·X (where X = x - 8/k), the directrix is X = -a, i.e., x = 8/k - k/4.
Setting this equal to x = 1: 8/k - k/4 = 1.
Multiply by 4k: 32 - k² = 4k → k² + 4k - 32 = 0 → (k+8)(k-4) = 0, giving k = 4 or k = -8.
The value among the options is k = 4.
For y² = 4a·X (where X = x - 8/k), the directrix is X = -a, i.e., x = 8/k - k/4.
Setting this equal to x = 1: 8/k - k/4 = 1.
Multiply by 4k: 32 - k² = 4k → k² + 4k - 32 = 0 → (k+8)(k-4) = 0, giving k = 4 or k = -8.
The value among the options is k = 4.
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