Question

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

Options

•  x² + y² = 32

• x² − y² = 16

• x² + y² = 16

•  x² − y² = 32

Answer 1

x² − y² = 32

The distance between the foci is  \[2ae\].

\[\therefore 2ae = 16\]

\[ \Rightarrow ae = 8\] \[e = \sqrt{2}\]

\[\therefore a\sqrt{2} = 8\]

\[ \Rightarrow a = 4\sqrt{2}\]

Also, \[b^2 = a^2 ( e^2 - 1)\]

\[ \Rightarrow b^2 = 32(2 - 1)\]

\[ \Rightarrow b^2 = 32\]

Standard form of the hyperbola is given below:

\[\frac{x^2}{32} - \frac{y^2}{32} = 1\]

\[ x^2 - y^2 = 32\]