Given equation of hyperbola is 16x2–9y2 = 576
i.e.,

16x2

576

−

9y2

576

= 1 ⇒

x2

36

−

y2

64

= 1 which is of the form

x2

a2

−

y2

b2

= 1.
The foci and vertices of the hyperbola lie on x-axis.
Now, a2 = 36 ⇒ a = 6 and b2 = 64 ⇒ b = 8
Also, c2 = a2+b2 = 36 + 64 = 100 ⇒ c = 10
∴ Coordinates of foci are (± c, 0) i.e, (±10, 0)
∴ Coordinates of vertices are (± a, 0) i.e. (± 6, 0)
Eccentricity (e) =

c

a

=

10

6

=

5

3

Length of latus rectum =

2b2

a

=

2×64

6

=

64

3

.