如图,抛物线y=x^2-2x-3于x轴交于a,b两点(a在b的左边),于y轴交于点c...

发布网友发布时间：2024-10-15 03:56

共1个回答

热心网友时间：2024-11-20 23:22

(1)

y = (x - 3)(x + 1)
A(-1, 0), B(3, 0), C(0, -3)
BC = 3√2
S = (1/2)*BC*BC上的高h = (1/2)*3√2*h = 6
h = 2√2
BC的方程: x/3 + y/(-3) = 1, x - y - 3 = 0
令E(e, e²- 2e - 3)
E与BC的距离h = 2√2 = |e - e² +2e + 3 - 3|/√2
|e² - 3e| = 4
e² - 3e + 4 = 0 (无解)
e² - 3e - 4 = (e - 4)(e + 1) = 0
e = -1, E(-1, 0), 与A重合
或
e = 4, E(4, 5)

(2)
G(0, g)

AOC的面积 = (1/2)*AO*CO = 3/2
GOB的面积 = (1/2)*OB*OG = (1/2)*3g = 3g/2 = 3/2
g = 1
BG: x/3 + y = 1
与抛物线联立, x = -4, F(-4, 21)
舍去x = 3 (点B)

(3)
G(g, g² - 2g - 3), 0 < g < 3
GOC的面积s1 = (1/2)*CO*g = 3g/2
GOB的面积s2 = (1/2)*OB*(-g² + 2g + 3) = (3/2)(-g² +2g +3)
3g/2 = 2*(3/2)(-g² +2g +3)
g = (3 + √57)/4 (另一解为负，舍去)
G((3 + √57)/4, -(3 + √57)/8)

(4)
N(n, n² - 2n - 3), 0 < n < 3
BC = 3√2
NBC = (1/2)BC*BC上的高h = (1/2)*3√2*h > 3
h > √2
BC的方程: x/3 + y/(-3) = 1, x - y - 3 = 0
N与BC的距离d = |n - n² +2n + 3 - 3|/√2 = |n² - 3n|/√2 >h = √2
|n² - 3n| >2
(i) n² - 3n > 2
n > (3 + √17)/2 >3, 舍去
或n < (3 - √17)/2 < 0, 舍去

(ii) n² - 3n < -2
(n - 1)(n - 2) < 0
1 < n < 2
存在