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Question
If the distance between the foci of a hyperbola is 16 and its eccentricity is `sqrt(2)`, then obtain the equation of the hyperbola.
Solution
Equation of hyperbola is `x^2/a^2 - y^2/b^2` = 1
Distance between the foci = 2ae
2ae = 16
⇒ ae = 8
⇒ `a xx sqrt(2)` = 8
⇒ `a = 8/sqrt(2) = 4sqrt(2)` ......`[∵ e = sqrt(2)]`
Now `b^2 = a^2(e^2 - 1)`
⇒ `b^2 - (4sqrt(2))^2 (2 - 1)`
⇒ b2 = 32
a = `4sqrt(2)`
⇒ a2 = 32
⇒ x2 – y2 = 32
Hence, the required equation is `x^2/32 - y^2/32` = 1
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