A hyperbola having the transverse axis of length $\sqrt 2$ has the same foci as that of the ellipse 3x² + 4y² = 12, then this hyperbola does not pass through which of the following points?

Ellipse : ${{{x^2}} \over 4} + {{{y^2}} \over 3} = 1$<br><br>eccentricity = $\sqrt {1 - {3 \over 4}} = {1 \over 2}$<br><br>$\therefore$ foci = ($\pm$ 1, 0)<br><br>for hyperbola, given $2a = \sqrt 2 \Rightarrow a = {1 \over {\sqrt 2 }}$<br><br>$\therefore$ hyperbola will be<br><br>${{{x^2}} \over {1/2}} - {{{y^2}} \over {{b^2}}} = 1$<br><br>eccentricity = $\sqrt {1 + 2{b^2}}$<br><br>$\therefore$ foci $= \left( { \pm \sqrt {{{1 + 2{b^2}} \over 2}} ,0} \right)$<br><br>$\because$ Ellipse and hyperbola have same foci<br><br>$\Rightarrow \sqrt {{{1 + 2{b^2}} \over 2}} = 1$<br><br>$\Rightarrow {b^2} = {1 \over 2}$<br><br>$\therefore$ Equation of hyperbola : ${{{x^2}} \over {1/2}} - {{{y^2}} \over {1/2}} = 1$<br><br>$\Rightarrow$ ${x^2} - {y^2} = {1 \over 2}$<br><br>Clearly $\left( {\sqrt {{3 \over 2}} ,{1 \over {\sqrt 2 }}} \right)$ does not lie on it.