Question

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

• A. $(\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}})$
• B. $(1,-\frac{1}{\sqrt{2}})$
• C. $(\frac{1}{\sqrt{2}},0)$
• D. $(-\sqrt{\frac{3}{2}},1)$

Answer 1

The correct option is A $(\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}})$

Ellipse: $\frac{x^2}{4}+\frac{y^2}{3}=1$

$b_1^2=a_1^2(1-e_1^2)$
$\Rightarrow 3=4(1-e_1^2)$
$\Rightarrow e_1=\frac{1}{2}$
Focus = $(\pm a_1{e}_1,0)$
$=(\pm 1,0)$

For hyperbola:
Length of transverse axis $2a_2=\sqrt{2}\Rightarrow a_2=\frac{1}{\sqrt{2}}$
Given, foci of hyperbola $=$ foci of ellipse
i.e. $a_2{e}_2=1$
$\Rightarrow e_2=\sqrt{2}$

$b_2^2=a_2^2(e_2^2-1)$
$\Rightarrow b_2^2=\frac{1}{2}(2-1)=\frac{1}{2}$
Therefore, equation of Hyperbola
$x^2-y^2=\frac{1}{2}$
Clearly, $(\sqrt{\frac{3}{2}},\frac{1}{\sqrt{2}})$ does not lie on hyperbola.