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Find the length of the conjugate axis of the hyperbola \(\frac{{{y^2}}}{{{16}}} - \frac{{{x^2}}}{{{49}}} = 1\) ?
1. 8
2. 14
3. 12
4. 10
5. None of these

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Correct Answer - Option 2 : 14

CONCEPT:

The properties of a vertical hyperbola \(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1\) are:

  • Its centre is given by: (0, 0)
  • Its foci are given by: (0, - ae) and (0, ae)
  • Its vertices are given by: (0, - a)  and (0, a)
  • Its eccentricity is given by: \(e = \frac{{\sqrt {{a^2} + {b^2}} }}{a}\)
  • Length of transverse axis = 2a and its equation is x = 0.
  • Length of conjugate axis = 2b and its equation is y = 0.
  • Length of its latus rectum is given by: \(\frac{2b^2}{a}\)


CALCULATION:

Given: Equation of hyperbola is \(\frac{{{y^2}}}{{{16}}} - \frac{{{x^2}}}{{{49}}} = 1\)

As we can see that, the given hyperbola is a vertical hyperbola.

So, by comparing the given equation of hyperbola with \(\frac{{{y^2}}}{{{a^2}}} - \frac{{{x^2}}}{{{b^2}}} = 1\) we get,

⇒ a2 = 16 and b2 = 49

As we know that, length of conjugate axis of a hyperbola is given by 2b

So, the length of the conjugate axis for the given hyperbola is: 2 × 7 = 14 units.

Hence, option B is the correct answer.

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