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

Answer 1

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`