Let ABCD be a rectangle with $A (0, 0), B (4, 0), C (4, 4) a n d D (0, 4)$ . Rectangle is folded in such a way that corner B always lies on line AD, then which of the following statement(s) is/are CORRECT?

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As the point B moves on AD the crease thus formed will touch a fixed parabola whose focus is at (4, 0).

As the point B moves on AD the crease thus formed will touch a fixed parabola whose focus is at $(\frac{3}{2}, 0)$ .

As the point B moves on AD the crease thus formed will touch a fixed parabola whose equation is $y^{2} = 8 (x - 2)$ .

As the point B moves on AD the crease thus formed will touch a fixed parabola whose equation of directrix is $y = 2 x .$

answer is A, C.

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Detailed Solution

$\cos (π - 2 θ) = \frac{4 - α}{α} α - 2 = 2 \cot^{2} θ$

Equation of crease is $y - 0 = \tan θ (x - (4 - α))$

$y = (x - 2) \tan θ + 2 \cot θ c = \frac{a}{m} \Rightarrow 2 \cot θ = \frac{a}{\tan θ} \Rightarrow a = 2$

$\Rightarrow y^{2} = 8 (x - 2)$ is equation of parabola.

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