A hyperbola having the transverse axis


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?

  1. (1) $\left(\frac{1}{\sqrt{2}}, 0\right)$

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

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

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

Correct Option: , 4


The given ellipse :


$\because c=\sqrt{a^{2}-b^{2}}=\sqrt{4-3}=1$

$\therefore$ Foci $=(\pm 1,0)$

Now for hyperbola :

Given : $2 a=\sqrt{2} \Rightarrow a=\frac{1}{\sqrt{2}}$

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

So, equation of hyperbola is

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

So, option (d) does not satisfy it.


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