Question

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

9y² – 4x² = 36

Answer 1

Equation of hyperbola 9y² – 4x² = 36

Dividing by 36, ${\displaystyle \frac{y^2}{4}-\frac{x^2}{9}=1}$

⇒ Transverse axis is along the x-axis.

∴ a² = 4, b² = 9

c² = a² + b² = 4 + 9 = 13

∴ a = 2, b = 3, c = ${\displaystyle \sqrt{13}}$

The coordinates of the vertices are (0, ±a) or (0, ±2)

The coordinates of the foci are (0, ±c) or ${\displaystyle (0,\pm \sqrt{13})}$

Eccentricity e = ${\displaystyle \frac{c}{a}=\frac{\sqrt{13}}{2}}$

Length of the latus rectum = ${\displaystyle \frac{2b^2}{a}}$

= ${\displaystyle \frac{2\times 9}{2}}$

= 9