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Question
Prove that the length of the latus rectum of the hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1 is `(2"b"^2)/"a"`
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Solution
The latus rectum LL’ of an hyperbola `x^2/"a"^2 - y^2/"b"^2` = 1 passes through S(ae, 0)
Hence L is (ae, y1)
`("a"^2"e"^2)/"a"^2 - y_1^2/"b"^2` = 1
`"e"^2 - 1 = y_1^2/"b"^2`
`y_1^2 = "b"^2("e"^2 - 1)`
= `"b"^2(1 + "b"^2/"a"^2 - 1) (because "e"^2 = 1 + "b"^2/"a"^2)`
`y_1^2 = "b"^4/"a"^2`
`y_1 = +- "b"^2/"a"`
End points of latus rectums are `("ae", "b"^2/"a")` and `("ae", - "b"^2/"a")`
∴ LL' = `"b"^2/"a" + "b"^2/"a"`
LL' = `(2"b"^2)/"a"`
Hence proved.
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