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

Toolbox:
Given $\;5y^2-9x^2=36.$ Dividing by $36 \rightarrow \large\frac{5}{36}$$y^2-\large\frac{9}{36}$$x^2=1 \rightarrow \large\frac{y^2}{\frac{6}{\sqrt5}} $$-\large\frac{x^2}{2}$$=1$
On comparing the given equation with the standard equation of a hyperbola, we get:
$\quad a =\large\frac{6}{\sqrt 5}$$, b = 2 Since a^2+b^2 = c^2 \rightarrow c = \sqrt{\frac{36}{5}+4}$$= \sqrt {\large\frac{56}{5}}$ $= 2\large\sqrt{\frac{14}{5}}$
1. Coordinates of Foci $= (0, \pm 2\large\sqrt{\frac{14}{5}}$$) 2. Coordinates of Vertices = (0,\pm \large\frac{6}{\sqrt 5}$$)$