Find the range of c for which the line $y = m x + c$ touches the parabola $y^{2} = 8 (x + 2) .$

(- \infty, - 4) \cup (4, \infty)

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(- \infty, - 2 \cup [2, \infty)

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(- \infty, - 4] \cup [4, \infty)

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(- \infty, - 4]

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Solution

The correct option is C $(- \infty, - 4] \cup [4, \infty)$
$\begin{matrix} I f e q u a t i o n o f tan g e n t, Y = m (X) + \frac{a}{m} f o r Y^{2} = 4 a X h e r e, y^{2} = 8 (x + 2) ∴ Y = y X = x + 2 y = m (x + 2) + \frac{2}{m} = m x + 2 m + \frac{2}{m} ∴ c = 2 m + \frac{2}{m} m \in R = 2 (m + \frac{1}{m}) ∴ c \geq 4 o r c \leq - 4 - 4 \geq c \geq 4 i f t + \frac{1}{t} \geq 2 f o r t \geq o r t + \frac{1}{t} \leq - 2 f o r t \leq 0 \end{matrix}$

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