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 - Mathematics and Statistics

Sum

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

The equation of the hyperbola is 16x2 – 9y2 = 144

∴ x^2/9 - y^2/16 = 1

Companng with x^2/"a"^2 - y^2/"b"^2 = 1, we get,

a2 = 9, 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) ae = 5(sqrt(5)/5) = sqrt(5)

Coordinates of foci= (± ae, 0) = (± 5, 0)

(5) "a"/"e" = 5/((sqrt(5)/5)) = 9/5

The equations of directrices are

x = ± "a"/"e" i.e. x = ± 9/5

(6) Length of latus rectum = (2"b"^2)/"a"

= (2(16))/3

= 32/3

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