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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:
16x2 – 9y2 = 144
Solution
Given equation of the hyperbola is 16x2 – 9y2 = 144
∴ `x^2/9 - y^2/16` = 1
Comparing this equation with `x^2/"a"^2 - y^2/"b"^2` = 1, we get,
a2 = 9 and b2 = 16
a = 3 and b = 4
(1) Length of transverse axis = 2a = 2(3) = 6
(2) Length of conjugate axis = 2b = 2(4) = 8
(3) Eccentricity = e = `sqrt("a"^2 + "b"^2)/"a"`
= `sqrt(9 + 16)/3`
= `sqrt(25)/3`
= `5/3`
(4) Co-ordinates of foci are S(ae, 0) and S'(−ae, 0),
i.e., `"S"(3(5/3),0)` and `"S'"(-3(5/3),0)`,
i.e., S(5, 0) and S'(−5, 0)
(5) Equations of the directrices are x = `±"a"/"e"`
∴ x = `± 3/((5/3))`
∴ x = `± 9/5`
(6) Length of latus rectum = `(2"b"^2)/"a"`
= `(2(16))/3`
= `32/3`
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