Correct Answer - Option 3 :

\(\frac{{{y^2}}}{{{36}}} - \frac{{{x^2}}}{{{64}}} = 1\)
__Concept__**:**

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

- Centre is given by: (0, 0)
- Vertices are given by: (0, ± a)
- Foci are given by: (0, ± c)
- 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 vertices and eccentricity of hyperbola are: (0, ± 6) and 5/3 respectively.

⇒ e = 5/3

∵ The vertices of the given hyperbola are of the form (0, ± a), it is a vertical hyperbola i.e it is of the form: \(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1\)

As we that the vertices of the vertical parabola i.e \(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1\) are of the form (0, ± a).

⇒ a = 6 and a^{2} = 36

As we know that eccentricity e = c/a

⇒ c = e ⋅ a

By substituting e = 5/3 and a = 6 in the above equation we get

⇒ c = 10 and c^{2} = 100

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

⇒ b2 = 100 - 36 = 64

Hence the equation of required hyperbola is: \(\frac{{{y^2}}}{{{36}}} - \frac{{{x^2}}}{{{64}}} = 1\)