Question

Equation of hyperbola whose axes are parallel to coordinate axis with $e=\sqrt{2}$ and having distance between the foci as $1$ unit is :

• A. $y^2-x^2=\frac{1}{8}$
• B. $x^2-y^2=5$
• C. $y^2-x^2=5$
• D. $x^2-y^2=\frac{1}{8}$

Answer 1

Answer: (A), (D)

The correct options are
A $y^2-x^2=\frac{1}{8}$
D $x^2-y^2=\frac{1}{8}$

Given,
If transverse axis along $x$ axis and conjugate along $y$ axis
$e=\sqrt{2}$ and $2ae=1$
$\Rightarrow a=\frac{1}{2\sqrt{2}}$
$\Rightarrow b^2=a^2(e^2-1)$
$\Rightarrow b^2=\frac{1}{8}(2-1)$
$\Rightarrow b^2=\frac{1}{8}$
Then the equation of hyperbola is
$\frac{x^2}{\frac{1}{8}}-\frac{y^2}{\frac{1}{8}}=1$
$\Rightarrow x^2-y^2=\frac{1}{8}$
If transverse axis is along $y$-axis and conjugate axis is along $x$ axis, then equation of hyperbola is
$y^2-x^2=\frac{1}{8}$