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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:
`y^2/25 - x^2/9` = 1
Solution
The equation of the hyperbola is `y^2/25 - x^2/9` = 1
Comparing with `y^2/"b"^2 - x^2/"a"^2` = 1, we get,
b2 = 25, a2 = 9
∴ b = 5, a = 3
(1) Length of transverse axis = 2b = 2(5) = 10
(2) Length of conjugate axis = 2a = 2(3) = 6
(3) Eccentricity = e = `sqrt("a"^2 + "b"^2)/"b"`
= `sqrt(25 + 9)/5`
= `sqrt(34)/5`
(4) be = `5(sqrt(34)/5) = sqrt(34)`
Coordinates of foci = (0, ± be) = `(0, ±sqrt(34))`
(5) `"b"/"e" = 5/((sqrt(34)/5)) = 25/sqrt(34)`
The equations of directrices are
y = `± "b"/"e"` i.e. y = `± 25/sqrt(34)`
(6) Length of latus rectum = `(2"a"^2)/"b"`
= `(2(9))/5`
= `18/5`
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