已知椭圆C：x^2/4+y^2/3=1,直线x=m上一点P与椭圆长轴的左右两端点A1,A2的连线分别交椭圆于另外一点M,

椭圆C：x^2/4+y^2/3=1①长轴的左右两端点为A1(-2,0),A2(2,0),
设P(m,p),p≠0,则PA1的斜率=p/(m+2),PA1:y=p(x+2)/(m+2),②
代入①,3x^2+4p^2*(x+2)^2/(m+2)^2=12,
3(x+2)(x-2)+4p^2*(x+2)^2/(m+2)^2=0,
x1=-2,或3(x-2)(m+2)^2+4p^2(x+2)=0,
[3(m+2)^2+4p^2]x=6(m+2)^2-8p^2,
xM=[6(m+2)^2-8p^2]/[3(m+2)^2+4p^2],
代入②,yM=12p(m+2)/[3(m+2)^2+4p^2].
同理,xN=[6(m-2)^2-8p^2]/[3(m-2)^2+4p^2],yN=12p(m-2)/[3(m-2)^2+4p^2],
以MN为直径的圆过过点A2,
向量A2M*A2N=0,
(xM-2)(xN-2)+yMyN=0,
(-16p^2)^2+12p(m+2)*12p(m-2)=0,
16+9(m^2-4)=0,9m^2=20,m^2=20/9,
∴m=土2√5/3.