Question

Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in  the distance between the foci = 16 and eccentricity = \[\sqrt{2}\].

Answer 1

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

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

\[ \Rightarrow b^2 = 32\]

Therefore, the standard form of the hyperbola is given below:

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

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