题目内容
已知椭圆C:
=1,过点M(2,0)且斜率不为0的直线交椭圆C于A,B两点.在x轴上若存在定点P,使PM平分∠APB,则P的坐标为________.
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设A(x1,y1),B(x2,y2),直线AB的方程为x=my+2.将直线AB的方程与椭圆C的方程联立,消去x得(4m2+9)y2+16my-20=0,所以y1+y2=
,y1y2=
.
若PM平分∠APB,则直线PA,PB的倾斜角互补,所以kPA+kPB=0.设P(a,0),则有
+
=0,将x1=my1+2,x2=my2+2代入上式,整理得
=0,
所以2my1y2+(2-a)(y1+y2)=0.将y1+y2=
,y1y2=
代入上式,整理得(-2a+9)·m=0.由于上式对任意实数m都成立,所以a=
.
综上,x轴上存在定点P
,使PM平分∠APB.
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若PM平分∠APB,则直线PA,PB的倾斜角互补,所以kPA+kPB=0.设P(a,0),则有
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所以2my1y2+(2-a)(y1+y2)=0.将y1+y2=
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综上,x轴上存在定点P
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