Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 5y² - 9x² = 36

Solution:

The given equation is 5y² - 9x² = 36

It can be written as

5y² - 9x² = 36

y²/(36/5) - x²/4 = 1

y²/(6/√5)² - x²/2² = 1 ....(1)

On comparing this equation with the standard equation of hyperbola

i.e., x²/a² + y²/b² = 1, we obtain

a = 6/√5 and b = 2.

We know that, c² = a² + b²

Hence,

⇒ c² = (6/√5)² + 2²

⇒ c² = 36/5 + 4

⇒ c² = 56/5

⇒ c = √56/5

⇒ c = 2√14/√5

Therefore,

The coordinates of the foci are (0, ± 2√14/√5)

The coordinates of the vertices are (0, ± 6/√5)

Eccentricity, e = c/a = (2√14/√5)/(6/√5) = √14/3

Length of latus rectum = 2b²/a = (2 × 4)/(6/√5) = (4√5)/3

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NCERT Solutions Class 11 Maths Chapter 11 Exercise 11.4 Question 5

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 5y² - 9x² = 36.

Summary:

The coordinates of the foci and vertices of the hyperbola 5y² - 9x² = 36 are (0, ± 2√14/√5), (0, ± 6/√5) respectively. The length of the latus rectum is (4√5)/3