Given:

The equation is 16x^{2} – 9y^{2} = 576

Let us divide the whole equation by 576, we get

\(\dfrac{ 16x^2}{576} –\dfrac{ 9y^2}{576} = \dfrac{576}{576}\)

\( \dfrac{x^2}{36} – \dfrac{y^2}{64 }= 1\)

On comparing this equation with the standard equation of hyperbola \( \dfrac{ x^2}{a^2} – \dfrac{y^2}{b^2} = 1\)

We get a = 6 and b = 8,

It is known that, a^{2} + b^{2} = c^{2}

So,

c^{2} = 36 + 64

c^{2} = \( \sqrt{100}\)

c = 10

Then,

The coordinates of the foci are (10, 0) and (-10, 0).

The coordinates of the vertices are (6, 0) and (-6, 0).

Eccentricity, e =\( \dfrac{c}{a} \)= \( \dfrac{10}{6} = \dfrac{5}{3} \)

Length of latus rectum =\( \dfrac{ 2b^2}{a} =\dfrac{ (2 × 8^2)}{6}\)

= \(\dfrac{ (2×64)}{6} = \dfrac{64}{3}\)

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