设椭圆方程：(x^2/a^2) + (y^2/b^2) = 1,变为：b^2x^2 + a^2y^2 = a^2b^2
∵e = 1/2 = c/a ,∴a^2 = 4c^2 = b^2 + c^2 ,∴a^2 = 4b^2/3
∴3b^2x^2 + 4b^2y^2 = 4b^2·b^2 ,令b^2 = t > 0 ,则3t·x^2 + 4t·y^2 = 4t^2 ,
将直线L方程代入可得：3t·x^2 + t·(x - 4)^2 = 4t^2
整理可得：4t·x^2 - 8tx + 16t - 4t^2 = 0 ,∵直线L是椭圆的切线,∴方程有相等实根,∴△ = 0 ,
∴整理得：4 = 4·1·(4 - t) ,∴t = 3 ,∴b^2 = 3 ,a^2 = 4
∴椭圆C的标准方程：(x^2/4) + (y^2/3) = 1