# A hyperbola having the transverse axis of length $\sqrt{2}$ has the same foci as that of the ellipse $3x^{2}+4y^{2}=12$, then this hyperbola does not pass through which of the following points? Option: 1 Option: 2 Option: 3 Option: 4

Given equation of ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$

$a^{2}=4, b^{2}=3\therefore e=\sqrt{1-\frac{3}{4}}=\frac{1}{2}$

Thus foci of ellipse $(\pm 1,0)$

$\Rightarrow \pm 1=\sqrt{a^{2}+b^{2}}\Rightarrow b^{2}=\frac{1}{2}$

or $b^{2}=1-a^{2}=1-\frac{1}{2}=\frac{1}{2}$

Now the equation of hyperbola

$\frac{x^{2}}{\frac{1}{2}}-\frac{y^{2}}{\frac{1}{2}}=1\Rightarrow 2x^{2}-2y^{2}=1$

Point $\left ( \sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}} \right )$

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