Question

初中数学题目请看图

请看图

Answer 1

(1)
M(2, 1)与N关于y = x对称, 则N(1, 2)
M为抛物线顶点，于是抛物线看表达为y = a(x - 2)² + 1
代入N的坐标，得2 = a(1 - 2)² + 1, a = 1
y = (x - 2)² + 1 = x² - 4x + 5

(2)
C(0, 5), MK为y = x/2
令K(k, k/2), CK的中点为D(k/2, (5+k/2)/2), 即D(k/2, (k + 10)/4), D在对称轴x = 2右侧，则k>4
D在抛物线上: (k + 10)/4 = (k/2 - 2)² + 1
k² - 9k + 10 = 0
k = (9 + √41)/2   （另一根小于4, 舍去）
D((9 + √41)/4, (29 + √41)/8)

(3)
令l的斜率为k, 则其方程为y - 2 = k(x - 1), y = kx + 2 - k
与抛物线联立: x² - 4x + 5 =kx + 2 - k
x² - (k + 4)x + k + 3 = (x - 1)(x - k - 3) = 0
x = k + 3, (x = 1为点N)
P(k+3, k² + 2k + 2)
PG = PF， 则∠PFG = ∠PGF，直线PG和PF倾斜角互补，PG的斜率为-k
-k = (k² + 2k + 2 - 5)/(k + 3 - 0)          (右边用C, P的坐标求斜率)
2k² + 5k - 3 = (2k - 1)(k + 3) = 0
k = 1/2, k = -3
请仔细看图中的蓝线(k = 1/2)和黑线(k = -3, 此时P和C重合)