Question:
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: , 4
Solution:
The given ellipse :
$\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$
$\because c=\sqrt{a^{2}-b^{2}}=\sqrt{4-3}=1$
$\therefore$ 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.