题目

椭圆C：x^2/a^2+y^2/b^2=1（a＞b＞0）的离心率为√6/3,短轴一个端点与两个焦点构成的S△=5√2/3
1,求椭圆方程 2,已知动直线y=k（x+1）与椭圆相交于A,B （1）若AB中点横坐标为负2分之1,求k （2）若点m（-7/3,0)求向量ma乘向量mb的值

提问时间：2020-10-16

答案

1.e² = 6/9 = 2/3 = c²/a²,a² = 3c²/2
S = (1/2)*2c*b = bc,= 5√2/3,b = 5√2/(3c)
a² = b² + c²
3c²/2 = [5√2/(3c)]² + c² = c² + 50/(9c²)
c² = 10/3
a² = 5,b² = 5/3
x²/5 + 3y²/5 = 1
2.
(1)
y = k(x + 1)
代入x²/5 + 3y²/5 = 1,(3k² + 1)x² + 6k²x + 3k² - 5 = 0
x₁ + x₂ = -6k²/(3k² + 1)
AB中点横坐标 = -1/2 = (x₁ + x₂)/2 = -3k²/(3k² + 1)
k² = 1/3
k = ±√3/3
(2)
由(1):x₁x₂ = (3k² - 5)/(3k² + 1)
y₁y₂ = k(x₁ + 1)*k(x₂ + 1) = k²(x₁x₂ + x₁ + x₂ + 1)
A(x₁,y₁),B(x₂,y₂)
向量MA = (x₁ + 7/3,y₁)
向量MB = (x₂ + 7/3,y₂)
向量MA∙向量MB = (x₁ + 7/3)(x₂ + 7/3) + y₁y₂
= x₁x₂ + (7/3)( x₁ + x₂) + 49/9 + k²(x₁x₂ + x₁ + x₂ + 1)
= (k² + 1)x₁x₂ + (7/3 + k²)(x₁ + x₂) + k² + 49/9
= (k² + 1)(3k² - 5)/(3k² + 1) + (7/3 + k²)( -6k²)/(3k² + 1) + k² + 49/9
= (k² + 1)(3k² - 5)/(3k² + 1) + (3k² + 7)( -2k²)/(3k² + 1) + k² + 49/9
= (3k⁴ - 2k² - 5 - 6k⁴ - 14k² + 3k⁴ + k²)/(3k² + 1) + 49/9
= (-15k² - 5)/(3k² + 1) + 49/9
= -5 +49/9
= 4/9