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 \left ( \frac{1}{\sqrt{2}},0 \right )
Option: 2 \left ( -\sqrt{\frac{3}{2}},1 \right )
Option: 3 \left ( 1,-\frac{1}{\sqrt{2}} \right )
Option: 4 \left ( \sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}} \right )

Answers (1)

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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