ISI2016-DCG-44

Question

If the distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2},$ then the equation of the hyperbola is

• A. $y^{2}-x^{2}=32$
• B. $x^{2}-y^{2}=16$
• C. $y^{2}-x^{2}=16$
• D. $x^{2}-y^{2}=32$

Answer 1

Assuming the focii lie on $x$-axis  and  the transverse axis is $y=0$  (i.e. $x-axis$).

So, distance between the two focii of the hyperbola is given by :  $2c=16$   $\Rightarrow$  $c=8$

Also, eccentricity of hyperbola is  $e=\frac{c}{a}$    $\Rightarrow$   $a=\frac{8}{\sqrt{2}}$    $\Rightarrow$    $a^{2}=32$

Also, for hyperbola  $c^{2}=a^{2}+b^{2}$    $\Rightarrow$    $b^{2}=64-\frac{64}{2}=32$

So, the equation of hypebola would be :   $\frac{x^{2}}{32}-\frac{y^{2}}{32}=1$    $\Rightarrow$    $x^{2}-y^{2}=32$

Option  D is correct.