作业帮求解答高三数学题 椭圆 如图
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求解答高三数学题 椭圆 如图
求解答高三数学题 椭圆 如图
求解答高三数学题 椭圆 如图
(1) c = √2
(b - 0)² + (0 - c)² = 3,b² + 2 = 3,b = 1
a² = b² + c² = 3
椭圆:x²/3 + y² = 1
a² + b² = 4
伴随圆:x² + y² = 4
(2)
设直线斜率为k,方程为y = kx + m,kx - y + m = 0
圆所截弦长2√2,半弦长l = √2 ,半径r = 2
圆心与直线距离h = |m|/√(k² + 1) = √(r² - l²) = √(4 - 2) = √2
m = 2(k² + 1) (i)
将直线方程代入x²/3 + y² = 1,整理得(3k² + 1)x² + 6kmx + 3(m² - 1) = 0
∆ = (6km)² - 12(3k² + 1)(m² - 1) = 12(3k² - m² + 1) = 0 (ii)
联立(i)(ii):m = -2 (舍去m = 2 > 0)
(3)
设Q(u,v),u² + v² = 4 (i)
过Q的切线斜率为k,方程:y - v = k(x - u),y = kx + v - ku (ii)
将(ii)代入x²/3 + y² = 1,整理得(3k² + 1)x² + 6k(v- ku)x + 3(v - ku)² - 3 = 0
∆ = [6km(v - ku)]² - 12(3k² + 1)[(v - ku) ² - 1]
= 36k² - 12(v - ku)² + 12 = 0
(3 - u²)k² + 2uvk + 1 - v² = 0
二直线斜率之积为k1*k2 = (1 - v²)/(3 - u²) = [1 - (4 - u²)]/(3 - u²) = -1