the proof of the eccentricity of an ellipse independent from the directrix, 
the eccentricity is defined as follows: 
For a given ellipse: 


*

*the length of the semi-major axis = $a$

*the length of the semi-minor = $b$

*the distance between the foci = $2c$ 

*the eccentricity is defined to be $\dfrac{c}{a}$ 
now the relation for eccenricity value in my textbook is $\sqrt{1- \dfrac{b^{2}}{a^{2}}}$
which I cannot prove. 

Consider an ellipse with center at the origin of course the foci will be at $(0,\pm{c})$ or $(\pm{c}, 0) $
As you have stated the eccentricity $e$=$\frac{c} {a}$
Note also that $c^2=a^2-b^2$, $c=\sqrt{a^2-b^2} $ where $a$ and $b$ are length of the semi major and semi minor axis and interchangeably depending on the nature of the ellipse
$e=\frac{c} {a}$ =$\frac{\sqrt{a^2-b^2}} {a}$=$\frac{\sqrt{a^2-b^2}} {\sqrt{a^2}}$
$e=\sqrt{\frac{a^2-b^2} {a^2} }$
Can you finish it from there?
