Answer
$$\frac{{{x^2}}}{4} - \frac{{{y^2}}}{{4096}} = 1$$
Work Step by Step
$$\eqalign{
& {\text{Vertices }}\left( { \pm 2,0} \right) \cr
& {\text{Asymptotes: }}y = \pm 32x \cr
& y{\text{ is constant, so the equation of the hyperbola is }} \cr
& \frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1 \cr
& {\text{With vertices }}\left( { \pm a,0} \right) \cr
& \left( { \pm 2,0} \right) \to a = 2 \cr
& {\text{The asymptores are:}} \cr
& y = \pm \frac{b}{a}x \cr
& \pm \frac{b}{a}x = \pm 32x \cr
& \frac{b}{a} = 32 \to b = 32a \cr
& b = 64 \cr
& {\text{Substituting the constants }}a{\text{ and }}b{\text{ into }}\left( {\bf{1}} \right) \cr
& \frac{{{x^2}}}{{{{\left( 2 \right)}^2}}} - \frac{{{y^2}}}{{{{\left( {64} \right)}^2}}} = 1 \cr
& \frac{{{x^2}}}{4} - \frac{{{y^2}}}{{4096}} = 1 \cr} $$
