• # 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)  $\sqrt{\frac{5}{3}}$

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)]