椭圆x^2/4+y^2/2=1上两个动点P(x1,y1)Q(x2,y2),且x1+x2=21>椭圆x^2/4+y^2/2=1上两个动点P(x1,y1)Q(x2,y2),且x1+x2=21>求证线段PQ的中垂线过一定点A2>设A关于原点o对称点为B,试求|PB|最小值及相应点P坐标

Question

椭圆x^2/4+y^2/2=1上两个动点P(x1,y1)Q(x2,y2),且x1+x2=2 1>
椭圆x^2/4+y^2/2=1上两个动点P(x1,y1)Q(x2,y2),且x1+x2=2
1>求证线段PQ的中垂线过一定点A
2>设A关于原点o对称点为B,试求|PB|最小值及相应点P坐标

Answer 1

1）x1^2/4+y1^2/2=1,①
x2^2/4+y2^2/2=1,②
①-②,(x1+x2)(x1-x2)/4+(y1+y2)(y1-y2)/2=0,
x1+x2=2,
∴(y1-y2)/(x1-x2)=-1/(y1+y2),
∴线段PQ的中垂线的斜率=y1+y2,方程为
y-(y1+y2)/2=(y1+y2)(x-1),
即y=(y1+y2)(x-1/2),它过定点A(1/2,0).
2)B(-1/2,0),
PB^2=(x1+1/2)^2+y1^2=x1^2+x1+1/4+2(1-x1^2/4)
=(1/2)x1^2+x1+9/4=(1/2)(x1+1)^2+7/4,
∴|PB|的最小值=√7/2,这时P(-1,土√6/2).