Correct Answer - Option 2 :

\(\frac{{{x^2}}}{{{16}}} - \frac{{{y^2}}}{{{9}}} = 1\)
__Concept__**:**

The hyperbola of the form \(\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\) has:

- Centre is given by: (0, 0)
- Vertices are given by: (± a, 0)
- Foci are given by: (± c, 0)
- Length of transverse axis is given by: 2a
- Length of conjugate axis is given by: 2b
- Eccentricity is given by:\(e = \sqrt {1 + \frac{{{b^2}}}{{{a^2}}}} \)
- b2 = c2 - a2

__Calculation__**:**

Given: The foci of hyperbola are: (± 5, 0) and the length of transverse axis is 8.

∵ foci lies on the x -axis so, it is horizontal hyperbola.

As we know that for the hyperbola of the form \(\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1\) .

The foci are given by: (± c, 0) and length of transverse axis is given by: 2a.

⇒ c = 5 and 2a = 8

⇒ a = 4, a^{2} = 16 and c^{2} = 25

As we know that b2 = c2 - a^{2}

By substituting the value of a^{2} and c^{2} in the above equation we get,

⇒ b2 = 25 - 16 = 9

Hence, the equation of required hyperbola is: \(\frac{{{x^2}}}{{{16}}} - \frac{{{y^2}}}{{{9}}} = 1\)