Answer
$$\frac{{{x^2}}}{{100}} - \frac{{{y^2}}}{{2500}} = 1$$
Work Step by Step
$$\eqalign{
& {\text{vertices at }}\left( { - 10,0} \right),\left( {10,0} \right);{\text{ asymptotes }}y = \pm 5x \cr
& {\text{The }}y{\text{ - coordinate in the vertices are the same, then the hyperbola}} \cr
& {\text{has the equation }}\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1 \cr
& {\text{With vertices at }}\left( { \pm a,0} \right){\text{ and asymptotes }}y = \pm \frac{b}{a}x \cr
& {\text{,Then}} \cr
& {\text{vertices }}\left( { - 10,0} \right),\left( {10,0} \right) \to \,\,\,a = 10 \cr
& y = \pm \frac{b}{a}x = \pm 5x \cr
& \frac{b}{a} = 5 \cr
& \frac{b}{{10}} = 5 \cr
& b = 50 \cr
& {b^2} = 2500 \cr
& {\text{The equation of the hyperbola is}} \cr
& \frac{{{x^2}}}{{100}} - \frac{{{y^2}}}{{2500}} = 1 \cr} $$
