AB是抛物线y=x^2上的点(异于原点),以AB为直径的圆经过原点,求证:直线AB经过定点
\u8fc7\u539f\u70b9\u7684\u76f4\u7ebfl\u4e0e\u629b\u7269\u7ebfy^2=4(x-1)\u4ea4\u4e8eA.B\u4e24\u70b9,\u4ee5AB\u4e3a\u76f4\u5f84\u7684\u5706\u6070\u597d\u8fc7\u7126\u70b9F\u8bbeL\uff1ay=kx(k\u22600)\u3002\u3002\u30021
y²=4(x-1)\u3002\u3002\u30022
\u8054\u7acb1\uff0c2\u5f97k²x²-4x+4=0
\u8bbeA(X1,Y1)B(X2,Y2)
\u7531\u6839\u4e0e\u7cfb\u6570\u7684\u5173\u7cfb\u6709\uff1a
x1+X2=4/k²\uff0cx1*x2=4/k²
\uff08x1-x2\uff09²=\uff08x1+X2\uff09²-4x1*x2
=16/k\u56db\u6b21\u65b9-16/k²\u3002\u3002\u3002\u30023
\u4ee3\u5165y=kx\u5f97\uff08y1-y2\uff09²=k²\uff08x1-x2\uff09²
=16/k²-16\u3002\u3002\u3002\u30024
|AB|²=\uff08x1-x2\uff09²+\uff08y1-y2\uff09²\u3002\u3002\u30025
\u7126\u70b9F(1\uff0c1)\uff0c\u5219\u5706\u7684\u534a\u5f84R²=1+1=2,
\u5219|AB|²=8\uff0c
\u8054\u7acb3\uff0c4\uff0c5\u89e3\u51fak
\u8bbeA(x1,y1),B(x2,y2)
y=-x+1
x=1-y
\u5219\uff1ay^2=2p(1-y)
y^2+2py-2p=0
y1+y2=-2p,y1y2=-2p
x1x2=(1-y1)(1-y2)=1-(y1+y2)+y1y2=1+2p-2p=1
\u4ee5\u5f26AB\u4e3a\u76f4\u5f84\u7684\u5706\u6070\u597d\u8fc7\u539f\u70b9
OA\u22a5OB
x1x2+y1y2=0
x1x2+y1y2=1-2p=0
p=1/2
\u629b\u7269\u7ebf\u7684\u65b9\u7a0b\u4e3a:y^2=x
消去y得:x^2-kx-b=0(△=k^2+4b≥0)
x1+x2=k,x1x2=-b,y1y2=(kx1+b)(kx2+b)=b^2
由于以AB为直径的圆经过原点,所以向量OA与向量OB的数量积为0
即 x1x2+y1y1=-b+b^2=0,所以b=1或b=0(舍)
于是直线AB为y=kx+1,过定点(0,1)
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设点A(a,a^2) B(b,b^2)
线段AB的中点C((a+b)/2,(a^2+b^2)/2)
因为AB为直径,且经过原点
则|OC|=|AB|/2
(a+b)^2/4+(a^2+b^2)^2/4=[(a-b)^2+(a^2-b^2)^2]/4
4ab+4a^2b^2=0
ab(ab+1)=0
因为点A、B异于原点
所以ab=-1
直线AB为:y-a^2=(a+b)(x-a)
(a+b)x-y-ab=0
(a+b)x-y+1=0
所以直线AB过定点(0,1)
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