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$The eccentricity of the hyperbola \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 is reciprocal to that of the ellipse$
$x^{2} + 4 y^{2} = 4 . If the hyperbola passes through a foci of the ellipse, then$

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a

the equation of the hyperbola is x23−y22=1

b

a focus of the hyperbola is (3,0)

c

the eccentricity of the hyperbola is 53

d

the equation of the hyperbola is x2−3y2=3

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detailed solution

Correct option is D

Given ellipse is x24+y21=1 . Eccentricity of the ellipse =1−14=32 Focus of the ellipse =(−3,0) Eccentricity of the hyperbola =1+b2a2 So, 1+b2a2=23⇒ba=13 Since the hyperbola passes through the focus of the ellipse, 3a2−0=1⇒a2=3 and b2=1 Thus, the equation of the hyperbola is x23−y21=1 or x2−3y2=3 Focus of hyperbola is (±2,0) .

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