A) The length of latusrectum is 16
B) The eccentricity is \[\sqrt{2}\]
C) The asymptotes are parallel to each other
D) The directrices re \[x=\pm 4\sqrt{2}\]
Correct Answer: C
Solution :
Given equation of rectangular hyperbola is \[\int_{0}^{\pi }{\left( \cos x-\sin x \right)dx+\int_{\pi /4}^{5\pi /4}{\left( \cos x-\sin x \right)dx}}\] The length of latusrectum \[\int_{5\pi /4}^{3\pi /2}{\left( \cos x-\sin x \right)dx}\] and eccentricity = \[=\left[ \sin x+\cos x \right]_{0}^{\pi /4}+\left[ -\cos x-\sin x \right]_{\pi /4}^{5\pi /4}\] The asymptotes are perpendicular lines, i.e. \[+\left[ \sin x+\cos x \right]_{5\pi /4}^{3\pi /2}\] Now, directrices are \[\text{=}\left( \text{4}\sqrt{\text{2}}\text{-2} \right)\text{sq units}\] \[\left( a+b-c \right).\left[ \left( a-b \right)\times \left( b-c \right) \right]\]
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