It is known that the eccentricity of the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) is √ 3 / 2, and the distance from the straight line passing through two points a (a, 0) B (0, - b) to the origin is four fifths of the root sign 5. The standard equation of the ellipse is solved

It is known that the eccentricity of the ellipse x ^ 2 / A ^ 2 + y ^ 2 / b ^ 2 = 1 (a > b > 0) is √ 3 / 2, and the distance from the straight line passing through two points a (a, 0) B (0, - b) to the origin is four fifths of the root sign 5. The standard equation of the ellipse is solved

E = C / a = √ 3 / 2, so C & # 178; = (3 / 4) a & # 178;
b²=a²-c²=a²/4,
a=2b (1)
The linear equation passing through two points a (a, 0) B (0, - b) is x / A + Y / (- b) = 1, that is, BX ay AB = 0
The distance from the origin to the straight line d = | 0 + 0-ab | / √ (A & # 178; + B & # 178;) = AB / √ (A & # 178; + B & # 178;) = 4 √ 5 / 5
Substituting a = 2B, we get 2B & # 178; / √ (5b & # 178;) = 4 √ 5 / 5, and the solution is b = 2, a = 4
The standard equation of ellipse is: X & # 178 / 16 + Y & # 178 / 4 = 1