如圖,在平面直角坐標系中,拋物線y=ax²+bx+c的對稱軸爲直線x=-3/2,拋物線與x軸的交點爲A、B,

題目：

如圖,在平面直角坐標系中,拋物線y=ax²+bx+c的對稱軸爲直線x=-3/2,拋物線與x軸的交點爲A、B,
與y軸的交點爲c,拋物線的頂點爲M,直線MC的解析式是y=3\4x-2
(1)求頂點M的坐標（2）求拋物線的解析式（3）以線段AB爲直徑做圓P,判斷直線MC與圓P的位置關係,並證明你的結論

解答：

(1)頂點在對稱軸 x= -3/2上
MC的解析式是y= (3/4)x - 2
x = -3/2,y = -9/8 -2 = -25/8
M(-3/2,-25/8)
(2) y = ax²+bx+c = a[x + b/(2a)]²+ c -b^2/(4a)
對稱軸爲x = -b/(2a) = -3/2,b= 3a (a)
C(0,-2)
-2 = 0 + 0 +c
c = -2 (b)
頂點M縱坐標 c -b^2/(4a) = -25/8 (c)
(a)(b)(c):a = 1/2,b = 3/2
求拋物線的解析式:y = (1/2)x² + (3/2)x - 2
(3) y = (1/2)x² + (3/2)x - 2 = 0
(x+4)(x-1)= 0
A(-4,0),B(1,0)
半徑 = (1+4)/2 = 5/2
圓心P(-3/2,0)
直線MC的解析式是y= (3/4)x - 2,3x - 4y - 8 = 0
圓心和直線MC的距離:|3(-3/2) - 4*0 -8|/√(3²+4²) = (25/2)/5 = 5/2,等於半徑,直線MC與圓相切