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`
