Find the length of latus rectum of the hyperbola 5y² - 9x² = 36 ? 

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 5y2 - 9x2 = 36.

The given equation of hyperbola can be re-written as: \(\frac{{{y^2}}}{{{\frac{36}{5}}}} - \frac{{{x^2}}}{{{4}}} = 1\)

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 = 36/5 and b² = 4

As we know that, length of latus rectum of a hyperbola is given by \(\frac{2b^2}{a}\)

So, the length of latus rectum of given hyperbola is 4√5/3 units

Hence, option A is the correct answer.