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Question
Find the length of transverse axis, length of conjugate axis, the eccentricity, the co-ordinates of foci, equations of directrices and the length of latus rectum of the hyperbola:
`x^2/25 - y^2/16` = 1
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Solution
The equation of the hyperbola is `x^2/25 - y^2/16` = 1
Comparing with `x^2/"a"^2 - y^2/"b"^2` = 1, we get,
a2 = 25, b2 = 16
(1) Length of transverse axis = 2a = 2(5) = 10
(2) Length of conjugate axis = 2b = 2(4) = 8
(3) Eccentricity = e = `sqrt("a"^2 + "b"^2)/"a"`
= `sqrt(25 + 16)/5`
= `sqrt(41)/5`
(4) ae = `5(sqrt(41)/5) = sqrt(41)`
Co-ordinates of foci ≡ (± ae, 0) = `(± sqrt(41), 0)`
(5) `"a"/"e" = 5/((sqrt(41)/5)) = 25/sqrt(41)`
The equations of directrices are
x = `± "a"/"e"` i.e., x = `± 25/sqrt(41)`
(6) Length of latus rectum = `(2"b"^2)/"a"`
= `(2(16))/5`
= `32/5`
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