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Directrix

If the line x...

Question

If the line $x - 1 = 0$ is the directrix of the parabola $y^{2} - k x + 8 = 0,$ then $k$ can be

\frac{1}{8}

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- 8

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4

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\frac{1}{4}

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Solution

The correct options are
B $- 8$
C $4$
The given parabola is
$y^{2} = k x - 8 = k (x - \frac{8}{k})$
Above parabola is in the form of $Y^{2} = k X$ where $X = x - \frac{8}{k}$ and $Y = y$ .
Directrix of this parabola is $X = - \frac{k}{4}$
i.e., $x = - \frac{k}{4} + \frac{8}{k}$
This will coincide with $x = 1$ if
$\frac{8}{k} - \frac{k}{4} = 1 \Rightarrow 32 - k^{2} = 4 k \Rightarrow k^{2} + 4 k - 32 = 0$
$\Rightarrow (k + 8) (k - 4) = 0$
$\Rightarrow k = - 8, 4$

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