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 [2000]

Question

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 [2000]

Smart Achievers · Accepted Answer

(3)

The directrix of the parabola $y^{2} = 4 a (x - x_{1})$ is given by $x = x_{1} - a .$

Now given parabola is

$y^{2} = k x - 8 \Rightarrow y^{2} = k (x - \frac{8}{k})$

$∴$ $Directrix of parabola is x = \frac{8}{k} - \frac{k}{4};$

Now, $x = 1$ also coincides with $x = \frac{8}{k} - \frac{k}{4}$

On comparing, $\frac{8}{k} - \frac{k}{4} = 1$ $\Rightarrow k^{2} - 4 k - 32 = 0$

$∴$ $k = 4$

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