Answer
$${x^2} + \frac{{{y^2}}}{{100}} = 1$$
Work Step by Step
$$\eqalign{
& {\text{Vertices }}\left( {0, \pm 10} \right){\text{ passing through the point }}\left( {\sqrt 3 /2,5} \right) \cr
& {\text{The equation of the ellipse is given by: }}\frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1,{\text{ with}} \cr
& {\text{Vertices }}\left( {0, \pm a} \right),\,\,\,a = 10 \cr
& \frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{{\left( {10} \right)}^2}}} = 1 \cr
& \frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{100}} = 1 \cr
& {\text{passing through the point }}\left( {\sqrt 3 /2,5} \right) \cr
& \frac{{{{\left( {\sqrt 3 /2} \right)}^2}}}{{{b^2}}} + \frac{{{{\left( 5 \right)}^2}}}{{100}} = 1 \cr
& \frac{3}{{4{b^2}}} + \frac{{25}}{{100}} = 1 \cr
& \frac{3}{{4{b^2}}} = \frac{3}{4} \cr
& {b^2} = 1 \cr
& b = 1 \cr
& \cr
& {\text{Then, the equation is}} \cr
& \frac{{{x^2}}}{{{{\left( 1 \right)}^2}}} + \frac{{{y^2}}}{{100}} = 1 \cr
& {x^2} + \frac{{{y^2}}}{{100}} = 1 \cr
& \cr
& {\text{Graph}} \cr} $$
