Thank you for reporting, we will resolve it shortly
Q.
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?
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 $2 a =\sqrt{2}$
$\Rightarrow a =\frac{1}{\sqrt{2}}$
$\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.