已知点F(1,0)为椭圆C:x^2/a^2+y^2/b^2=1的右焦点,过点A(a,0)、B（0,b）的直线与圆x^2+y^2=1=7/121)求椭圆方程2）过点F的直线叫椭圆C与M、N两点,求证1/MF+1/NF为定值

Question

已知点F(1,0)为椭圆C:x^2/a^2+y^2/b^2=1的右焦点,过点A(a,0)、B（0,b）的直线与圆x^2+y^2=1=7/12
1)求椭圆方程
2）过点F的直线叫椭圆C与M、N两点,求证1/MF+1/NF为定值

Answer 1

l:y=√3(x-c),%D%A代入x^2/a^2+y^2/b^2=1得%D«^2x^2+3a^2(x^2-2cx+c^2)=a^2b^2,%D%A整理得(4a^2-c^2)x^2-6a^2cx+4a^2c^2-a^4=0,①%D¡)设A(x1,y1),B(x2,y2),则x1+x2=6a^2c/(4a^2-c^2),%D%A由向量AF=2FB得c-x1=2(x2-c),x1+2x2=3c,%D%A由焦半径公式,AF=a-ex1,FB=a-ex2,∴a-ex1=2(a-ex2),∴e(x1-2x2)=-a,∴x1-2x2=-a^2/c,%D%A解得x1=(3c-a^2/c)/2,x2=(3c+a^2/c)/4,%D%A∴x1+x2=(9c-a^2/c)/4=6a^2c/(4a^2-c^2),%D%A∴(9c^2-a^2)(4a^2-c^2)=24a^2c^2,%D%A两边都除以a^4,得(9e^2-1)(4-e^2)=24e^2,9e^4-13e^2+4=0,0