Let the tangent drawn to the parabola $y^{2} = 24 x$ at the point $(α, β)$ is perpendicular to the line $2 x + 2 y = 5$ . Then the normal to the hyperbola $\frac{x^{2}}{α^{2}} - \frac{y^{2}}{β^{2}} = 1$ at the point $(α + 4, β + 4)$ does not pass through the point :

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(30,8)

(20,12)

(15,13)

(25,10)

answer is D.

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Detailed Solution

$Slope of the tangent to y^{2} = 24 x at (α, β) = 1 β^{2} = 24 α, Equation of tangent y β = 12 (x + α), Slope = \frac{12}{β} = 1 \begin{array}{l} \Rightarrow α = 6, β = 12 \\ ∴ (α + 4, β + 4) = (10, 16) \\ Normal at (10, 16) to \frac{x^{2}}{36} - \frac{y^{2}}{144} = 1 is \frac{36 x}{10} + \frac{144 y}{16} =36+144 \\ \Rightarrow 2 x + 5 y = 100 \end{array}$