Write the equation of the hyperbola of eccentricity \[\sqrt{2}\], if it is known that the distance between its foci is 16.
Solution
The foci of the hyperbola are of the form \[\left( ae, 0 \right)\] and \[\left( - ae, 0 \right)\].
Distance between the foci = \[\sqrt{\left( ae - ( - ae \right)^2 + 0^{{}^2}}\]
\[ = \sqrt{\left( 2ae \right)^{{}^2}}\]
\[ = 2ae\]
Distance between the foci is 16 and eccentricity of the hyperbola is \[\sqrt{2}\].
\[\therefore 2ae = 16\]
\[ \Rightarrow 2\sqrt{2}a = 16\]
\[ \Rightarrow a = 4\sqrt{2}\]
Now, \[b^2 = a^2 ( e^2 - 1)\]
\[ \Rightarrow b^2 = \left( 4\sqrt{2} \right)^2 ((\sqrt{2} )^2 - 1)\]
\[ \Rightarrow b^2 = 32\]
Equation of the hyperbola is given below:
\[\frac{x^2}{\left( 4\sqrt{2} \right)^2} - \frac{y^2}{32} = 1\]
\[ \Rightarrow \frac{x^2}{32} - \frac{y^2}{32} = 1\]
