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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 = \[\sqrt{2}\].
Solution
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:
\[\frac{x^2}{32} - \frac{y^2}{32} = 1\]
\[ x^2 - y^2 = 32\]
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