Correct Answer - Option 2 :
\(\frac{{{y^2}}}{{{25}}} - \frac{{{x^2}}}{{{11}}} = 1\)
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: \(e = √ {1 + \frac{{{b^2}}}{{{a^2}}}} \)
- b2 = c2 - a2
Calculation:
Given: The foci and the length of conjugate axis are: (0, ± 6) and 2√11 respectively.
∵ The foci of the given hyperbola are of the form (0, ± c), it is a vertical hyperbola i.e it is of the form:
\(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1\)
⇒ c = 6 and c2 = 36.
As we know that he length of conjugate axis of the hyperbola of the form \(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1\) is given by: 2b
⇒ 2b = 2√11
⇒ b = √11 and b2 = 11.
As we know that, b2 = c2 - a2
⇒ a2 = c2 - b2 = 36 - 11 = 25
Hence, the equation of required hyperbola is: \(\frac{{{y^2}}}{{{25}}} - \frac{{{x^2}}}{{{11}}} = 1\)