1.C(0,1)
L2:y = 3
y = x²/2 -3x/2 + 1 =3
x² - 3x -4 = 0
(x-4)(x+1) = 0
A(-1,3),B(4,3)
AC的斜率k1 = (3-1)/(-1-0) = -2
BC的斜率k2 = (3 - 1)/(4 - 0) = 1/2
k1 * k2 = -1
ΔABC为直角三角形
2.l1:y = c
l2:y = c + t (t > 0)
l2与抛物线的交点为A、B,ax²+bx + c = c + t
ax²+bx - t =0
A,B的坐标为
A([-b - √(b²+4at)]/(2a),c + t)
B([-b + √(b²+4at)]/(2a),c + t)
C(0,c)
AC的斜率k1 = (c+t-c)(2a)/[-b - √(b²+4at)] = -2at/[b + √(b²+4at)]
BC的斜率k2 = (c+t-c)(2a)/[-b + √(b²+4at)] = 2at/[-b + √(b²+4at)]
k1*k2 = -1
4a²t² = 4at
at = 1
t = 1/a
3.抛物线的对称轴:x = -b/(2a)
A关于y轴的对称点A' 的坐标为([b + √(b²+4at)]/(2a),c + t)
|A'B| = [-b + √(b²+4at)]/(2a) - [b + √(b²+4at)]/(2a) = -b/a
a > 0,显然 b < 0
A' 在抛物线的对称轴上,-b/(2a) = [b + √(b²+4at)]/(2a)
-2b = √(b²+4at)
3b² = 4at = 4
b = -√(4/3) (正值舍去)
y = ax²+bx + c =c
ax²+bx = 0
x(ax +b) = 0
x = 0(点C),x = -a/b
D(-a/b,c)
|CD| = -a/b
AB,CD平行
|CD| = |A'B|,四边形A'CDB为平行四边形,CD上的高h = c+t -c = t
四边形A'CDB的面积S = |CD|*h = -(a/b)t = -at/b = -1/b = 1/√(4/3) = (√3)/2