A hyperbola having the transverse axis of length $\sqrt{2}$ has the same foci as that of the ellipse $3 x^{2}+4 y^{2}=12$, then this hyperbola does not pass through which of the following points ?
Correct Option: , 2
Ellipse : $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$
eccentricity $=\sqrt{1-\frac{3}{4}}=\frac{1}{2}$
$\therefore$ foci $=(\pm 1,0)$
for hyperbola, given $2 \mathrm{a}=\sqrt{2} \Rightarrow \mathrm{a}=\frac{1}{\sqrt{2}}$
$\therefore$ hyperbola will be
$\frac{x^{2}}{1 / 2}-\frac{y^{2}}{b^{2}}=1$
eccentricity $=\sqrt{1+2 b^{2}}$
$\therefore$ foci $=\left(\pm \sqrt{\frac{1+2 b^{2}}{2}}, 0\right)$
$\because$ Ellipse and hyperbola have same foci
$\Rightarrow \sqrt{\frac{1+2 b^{2}}{2}}=1$
$\Rightarrow \quad b^{2}=\frac{1}{2}$
$\therefore$ Equation of hyperbola : $\frac{x^{2}}{1 / 2}-\frac{y^{2}}{1 / 2}=1$
$\Rightarrow x^{2}-y^{2}=\frac{1}{2}$
Clearly $\left(\sqrt{\frac{3}{2}}, \frac{1}{\sqrt{2}}\right)$ does not lie on it.