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If chord, which is not a tangent, of the parabola $y^{2} = 16 x$ has the equation, $2 x + y = p$ and midpoint, $(h, k)$ the which of the following is (are) possible values (S) of $p, h, k$

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$p = 5, h = 4, k = - 3$

$p = 2, h = 3, k = - 4$

$p = - 1, h = 1, k = - 3$

$p = - 2, h = 2, k = - 4$

answer is A.

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Detailed Solution

$\begin{matrix} S_{1} = S_{11} \\ k y - 8 (x + h) = k^{2} - 16 h \\ k y - 8 x + 8 h - k^{2} = 0 \\ - k y + 8 x + k^{2} - 8 h = 0 \end{matrix}$

Comparing the given equation

$\begin{array}{l} y + 2 x - p = 0 \\ \frac{- k}{1} = 4 = \frac{k^{2} - 8 h}{- p} \\ \frac{- k}{1} = 4 = \frac{k^{2} - 8 h}{- p} \\ - 4 p = k^{2} - 8 h and 8 h - 4 p = 16 \\ 2 h - p = 4 \end{array}$

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