अतिपरवलयों के शीर्षों, नाभियों के निर्देशांक, उत्केंद्रता और नाभिलंब जीवा की लंबाई ज्ञात कीजिए

$49 y^{2}-16 x^{2}=784$

The given equation is $49 y^{2}-16 x^{2}=784$

It can be written as  $49 y^{2}-16 x^{2}=784$

Or,  $\frac{y^{2}}{16}-\frac{x^{2}}{49}=1$

Or,  $\frac{y^{2}}{4^{2}}-\frac{x^{2}}{7^{2}}=1$          ......... $(1)$

On comparing equation $(1)$ with the standard equation of hyperbola i.e., $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1,$ we obtain $a=4$ and $b=7$

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

$\therefore c^{2}=16+49=65$

$\Rightarrow c=\sqrt{65}$

Therefore, The coordinates of the foci are $(0,\,\pm \sqrt{65})$

The coordinates of the vertices are $(0,\,±4)$

Eccentricity, $e=\frac{c}{a}=\frac{\sqrt{65}}{4}$

Length of latus rectum $=\frac{2 b^{2}}{a}=\frac{2 \times 49}{4}=\frac{49}{2}$