# Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 5y^{2} - 9x^{2} = 36

**Solution:**

The given equation is 5y^{2} - 9x^{2} = 36

It can be written as

5y^{2} - 9x^{2} = 36

y^{2}/(36/5) - x^{2}/4 = 1

y^{2}/(6/√5)^{2} - x^{2}/2^{2} = 1 ....(1)

On comparing this equation with the standard equation of hyperbola

i.e., x^{2}/a^{2} + y^{2}/b^{2} = 1, we obtain

a = 6/√5 and b = 2.

We know that, c^{2} = a^{2} + b^{2}

Hence,

⇒ c^{2} = (6/√5)^{2} + 2^{2}

⇒ c^{2} = 36/5 + 4

⇒ c^{2} = 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 = 2b^{2}/a = (2 × 4)/(6/√5) = (4√5)/3

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 5y^{2} - 9x^{2} = 36.

**Summary:**

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