Question

一道椭圆题已知椭圆C的中心在坐标原点,长轴在X轴上,F1、F2分别为其左右焦点,P是椭圆上任意一点,且向量F1P点乘向量

Answer 1

1.x^2/a^2+y^2/b^2=1

F1(-c,0),F2(c,0),P(x0,y0)

F1P*F2P=(x0+c,y0)(x0-c,y0)

=x0^2+y0^2-c^2

=x0^2+b^2(1-x0^2/a^2)-c^2

=(c^2/a^2)x0^2+b^2-c^2

当x0^2=0,b^2-c^2=-2(题目打错了)

当x0^2=a^2,b^2=1,

c^2=3,a^2=4

x^2/4+y^2=1

2.椭圆方程与y=kx+m联立

得到(1+4k^2)x^2+8kmx+4m^2-4=0

x1+x2=-8km/(1+4k^2),x1x2=(4m^2-4)/(1+4k^2)

A(2,0),M(x1,y1),N(x2,y2)

AM*AN=(x1-2,y1)(x2-2,y2)

=x1x2-2(x1+x2)+4+y1y2=0

y1=kx1+m,y2=kx2+m

代入整理得

(k^2+1)x1x2+(km-2)(x1+x2)+m^2+4=0

代入x1+x2,x1x2

得5m^2+16km+12k^2=0

(5m+6k)(m+2k)=0

m=-2k,y=kx+m=k(x-2) 舍,

m=-6k/5,y=kx+m=k(x-6/5)

直线L过定点(6/5,0)