Question

椭圆2x^2+y^2=2,一直线过点P(0,m)与椭圆交于A,B不同两点,若有AP向量=n向量PB,向量oA+n向量oB

Answer 1

A(x1,y1),B(x2,y2)

AP直线：y=kx+m

y1+ny2=k(x1+nx2)+(1+n)m 1)

OA+nOB=4OP

x1+nx2=0,y1+ny2=4m 2)

比较1）2）4m=(1+n)m

n=3

AP=nPB=3PB

Px=(x1+3x2)/(1+3) Py=(y1+3y2)/(1+3)=m

x1=x2=0时

y^2=2,y1=√2,y2=-√2 m=(√2-3√2)/4=-√2/2

y1=-√2,y2=√2,m=(-√2+3√2)/4=√2/2

-√2/2