Obtain the equation of a hyperbola with coordinate axes as principal axes given that the distances of one of its vertices from the foci are 9 and I units.

Let equation of hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ 
If vertices are  $A(a,0)$  and  $A\text{'}(-a,0)$   and foci are  $S(ae,0)$ and  $S\text{'}(-ae,0)$
Given  $I(S\text{'}A)=9and\mathcal{l}\left(SA\right)=1$
 $\Rightarrow a+ae=9andae-a=1$
or  $a(1+e)=9$   and  $a(e-1)=1$
 $$\begin{gathered} \therefore \frac{a(1+e)}{a(e-1)}=\frac{9}{1} \\ \Rightarrow 1+e=9e-9\Rightarrow e=\frac{5}{4} \\ \because a(1+e)=9 \\ \therefore a(1+\frac{5}{4})=9\Rightarrow a=4 \\ {b}^2=a^2(e^2-1)=16(\frac{25}{16}-1) \\ \therefore b^2=9 \\ \frac{x^2}{16}-\frac{y^2}{9}=1 \end{gathered}$$