Answer 1

\[\frac{\sqrt{5}}{2}\]

The equation of the hyperbola is \[x^2 - 4 y^2 = 1\].

This can be rewritten in the following way:

\[\frac{x^2}{1} - \frac{y^2}{\frac{1}{4}} = 1\]

This is the standard form of a hyperbola, where 

\[a^2 = 1 \text { and }b^2 = \frac{1}{4}\]

The value of eccentricity is calculated in the following way:

\[b^2 = a^2 ( e^2 - 1)\]

\[ \Rightarrow \frac{1}{4} = ( e^2 - 1)\]

\[ \Rightarrow e^2 = \frac{5}{4}\]

\[ \Rightarrow e = \frac{\sqrt{5}}{2}\]