已知椭圆E：x^2/a^2+y^2/b^2=1(a>b>0)过点p(1,3/2),离心率e=1/2,右顶点为A,右焦点为F.(1)求椭圆E的标准方程；（2）若经过F的直线l（不与x轴重合）交椭圆E与B,C两点,延长BA,CA,分别交右准线于M,N两点.求证

Question

已知椭圆E：x^2/a^2+y^2/b^2=1(a>b>0)过点p(1,3/2),离心率e=1/2,右顶点为A,右焦点为F.(1)求椭圆E的标准方程；（2）若经过F的直线l（不与x轴重合）交椭圆E与B,C两点,延长BA,CA,分别交右准线于M,N两点.求证：FN
垂直于FM

Answer 1

(1)
过点P:1/a² + 9/(4b²) = 1 (i)
e² = c²/a² = (a² - b²)/a² = 1 - b²/a² = 1/4
3a² = 4b² (ii)
联立(i)(ii):a² = 4,b² = 3,c² = 1
x²/4 + y²/3 = 1
(2)
A(2,0),F(1,0)
右准线:x = a²/c = 4/1 = 4
设直线l的斜率为k,方程为 y = k(x - 1)
代入x²/4 + y²/3 = 1 并整理:(4k² + 3)x² - 8k²x + 4(k² - 3) = 0
x₁ + x₂= 8k²/(4k² + 3) (iii)
x₁x₂ = 4(k² - 3)/(4k² + 3) (iv)
B(x₁,k(x₁ - 1))
C(x₂,k(x₂ - 1))
AB的方程:(y - 0)/[k(x₁ - 1) - 0] = (x - 2)/(x₁ - 2)
取x = 4 (准线),M的纵坐标为y₁ = 2k(x₁ - 1)/(x₁ - 2)
类似可得N的纵坐标为y₂ = 2k(x₂ - 1)/(x₂ - 2)
FM的斜率k₁ = (y₁ - 0)/(4 - 1) = y₁/3
FN的斜率k₂ = (y₂ - 0)/(4 - 1) = y₂/3
k₁k₂ = (1/9)y₁y₂ = (4k²/9)(x₁ - 1)(x₂ - 1)/[(x₁ - 2)(x₂ - 2)]
= (4k²/9)[x₁x₂ - (x₁ + x₂) + 1]/[x₁x₂ - 2(x₁ + x₂) + 4]
代入(iii)(iv),可得k₁k₂ = -1
FN垂直于FM