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Question
Find the equation of the hyperbola satisfying the given conditions:
Foci `(+-3sqrt5, 0)`, the latus rectum is of length 8.
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Solution
Foci `(± 3sqrt5, 0)` the latus recum is of length 8.
Here, the foci are on the x-axis.
Therefore, the equation of the hyperbola is of the form `x^2/a^2 - y^2/b^2 = 1`
Since the foci are `(± 3sqrt5, 0)`, C = `±3sqrt5`
Length of latus retum = 8
`(2b^2)/a = 8`
= b2 = 4a
We know that a2 + b2 = c2
∴ a2 + 4a = 45
= a2 + 4a - 45 = 0
= a2 + 9a - 5a - 45 = 0
= (a + 9) (a - 5) = 0
= a = -9, 5
but a ≠ −9
Thus, the equation of the hyperbola is `x^2/25 - y^2/20 = 1`.
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