如图,抛物线y=ax平方+bx+c(a≠0)与x轴相交于点A(-1,0),B(3,0)两点,与y轴交于点C（0,-3）以

(1)
抛物线y=ax平方+bx+c(a≠0)与x轴相交于点A(-1,0),B(3,0)两点, 可表达为y = a(x + 1)(x - 3)
x = 0, y = -3a = -3 (点C)
a = 1
y = (x + 1)(x - 3) = x² - 2x - 3
对称轴x = (3 - 1)/2 = 1, 顶点(1, -4)
(2)
M(1, 0)
令E(-1, e), PD的斜率为k, 方程为y - e = k(x + 1), kx - y + k + e = 0 (i)
圆M的半径为2 = MD = |k - 0 + k + e|/√(k² + 1) = |2k + e|/√(k² + 1)
k = (4 - e²)/(4e)
MD的斜率为-1/k, 方程为y - 0 = (-1/k)(x - 1) (ii)
从(i)(ii)可得D((1 - ek - k²)/(k² + 1), (2k + e)/(k² + 1))
令D(u, v), u = (1 - ek - k²)/(k² + 1), v = (2k + e)/(k² + 1) (iii)
从D作AE的垂线,垂足F(-1, v)
四边形EAMD的面积S= 梯形AMDF的面积 + 三角形DEF的面积
= (1/2)(AM + FD)AF + (1/2)FD*FE
2S = 8√3 = (AM + FD)AF + FD*FE
= [2 + u -(-1)]*v + [u - (-1)]*(e - v)
= eu + 2v + e
代入u, v, k,得e = 2√3
k = -1/√3
PD方程: y - 2√3 = (-1/√3)(x + 1)
y = (5 - x)/√3
(3)
M为DN的中点, N(m, n)
1 = (u + m)/2, m = 2 - u
0 = (v + n)/2, n = -v
N(2 - u, -v)
接(2), 四边形EAMD的面积S = (eu + 2v + e)/2
△DAN的面积S' = (1/2)AM*|N的纵坐标|
= (1/2)*2*v = v
S = S', (eu + 2v + e)/2 = v
u = -1, 此为点A(-1, 0)
P点不存在