Question

Find the coordinates of foci, the vertices, eccentricity and the length of latus rectum of the following hyperbolas.

1. \(\frac{y^2}{9} - \frac{x^2}{27} = 1\)

2. 5y² - 9x² = 36

Answer 1

1. The equation of the hyperbola is of the form

\(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) ⇒ a² = 9; b² = 27

⇒ c² = a² + b² ⇒ c² = 9 + 27 = 36 ⇒ c = 6

Coordinate of foci are (0, ±6)

Coordinate of vertices are (0, ±a) ⇒ (0, ±3)

Eccentricity = \(\frac{c}{a} = \frac{6}{3}\)=2

Length of latus rectum = \(\frac{2b^2}{a} = \frac{2\times 27}{3} \)=18.

2. Given; 5y² – 9x² = 36

The equation of the hyperbola is of the form