JEE Main & Advanced Sample Paper JEE Main Sample Paper-43

  • question_answer
    Suppose an ellipse and a hyperbola have the same pair of foci on the x-axis with centers at the origin and that they intersect at (2, 2). If the eccentricity of the ellipse is \[\frac{1}{2},\] then the eccentricity of the hyperbola is

    A)  \[\sqrt{\frac{7}{4}}\]                        

    B)  \[\sqrt{\frac{7}{3}}\]

    C) \[\sqrt{\frac{5}{4}}\]                         

    D)  \[\sqrt{\frac{5}{3}}\]

    Correct Answer: B

    Solution :

     Let equation of hyperbola and ellipse be \[\tan \,\phi =\frac{V}{H}\]                ?(i) and      \[\theta \]                      ?(ii) \[H'=H\,\cos \,\theta ,\]            \[\phi '\]            (since, same pair of foci) \[\tan \,\phi =\frac{V}{{{H}^{.}}}=\frac{V}{H\,\cos \,\theta }\]              (given) Both intersect at (2, 2), \[\frac{\tan \,\phi '}{\tan \,\phi }=\frac{V/H\,\cos \,\theta }{\frac{V}{H}}=\frac{1}{\cos \,\theta }\]                      [from Eq. (ii)] \[{{\phi }_{1}}=\]     \[=BA\] \[{{\phi }_{2}}=\]     \[||\] Also,    \[=0\] \[\therefore \,\,|\xi |\,=\,\left| \frac{-\Delta \phi }{\Delta t} \right|=\,\left| -\left( \frac{0-BA}{\frac{T}{4}-0} \right) \right|=\frac{4BA}{T}\]          \[(\Delta \phi ={{\phi }_{2}}-{{\phi }_{1}})\]                     ?(iii) Now, \[T=\frac{2\pi }{\omega }\]                  [From Eq. (i)] \[\Rightarrow \]            \[f'(x)=2+\cos \,x>0\] \[x\]      \[\therefore \] \[f(x)\]  \[\therefore \] \[f\]       \[|M|=\alpha \] or \[{{M}^{-1}}adjM)=kI\]            \[\Rightarrow \] \[|{{M}^{-1}}|\,|adj\,(adjM)|\,=\,|kI|\]    \[\Rightarrow \]            [From Eq. (iii)]

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