Question

求解 要过程的！！！！分会高高的！！！

Answer 1

1.
(1)
圆的圆心为原点，半径r为2
直线被圆所截的半弦为d/2 = √3
设圆心到直线的距离为h (h >0): h² + (d/2)² = r², h² + 3 = 4, h = 1
y = kx + 2, kx - y + 2 = 0

h = |k*0 - 0 + 2|/√(k² + 1) = 2/√(k² + 1) = 1
k = √3 (舍去-√3, 因为直线斜率显然>0)

(2)
直线在y轴上的截距为2 = b
与(1)类似, d = 4√5/5时, k = 1/2 (舍去-1/2)
y = -x/2 + 2
y = 0, x = 4, c = 4
a = √(b² + c²) = √(4 + 16) = 2√5
e = c/a = 4/(2√5) = 2√5/5
d > 4√5/5时, 直线绕B向y轴移动, 焦点更靠近原点, 椭圆更接近圆, 即e ≤ 2√5/5

2.
(1) 显然∠A, ∠B均为锐角, 故只须∠AF2F1小于45˚, tan∠AF2F1 = F1A/F1F2 < 1
不妨设在第二象限, x = -c, c²/a² - y²/b² = 1
y = b²/a
A(-c, b²/a)
F1A = b²/a, F1F2 = 2c
tan∠AF2F1 = (b²/a)/(2c) = b²/(2ac) < 1
b² < 2ac (i)
e² = c²/a² = (a² + b²)/a² = 1 + b²/a² < 1 + 2ac/a² = 1 + 2e
e² - 2e - 1 < 0
1 - √2 < e < 1 + √2
C

(2)
a + b = 5
ab = (√6)² = 6
a² - 5a + 6 = 0
(a - 2)(a - 3) = 0
a =2, b = 3 (a < b, 舍去)
a = 3, b = 2
c = √(3*3 + 2*2) =√13
e = √13/3
D

(3)
b = 1
e = c/a = √[(a² - b²)/a²] = √(1 - b²/a²) = √(1 - 1/a²)
a > 4, e > √(1 - 1/4²) = √15/4
D