数学竞赛中的解析几何问题_一_

()(,310036):O12-4:A:0488-7395(2000)08-0039-03:2000-02-15.,,.1P1(x1,y1),P2(x2,y2),1)|P1P2|=(x1-x2)2+(y1-y2)2=1+k2|x1-x2|(k=y2-y1x2-x1P1P2);2)P(x,y)P1P2(-1),x=x1+x21+,y=y1+y21+;3)P1P2y-y1=k(x1-x2)(k),Ax+By+C=0(A,B,C);4)P1P23),y=kx+b(),xa+yb=1();5)P0(x0,y0),P0Ax+By+C=0d=|Ax0+By0+C|A2+B2;6)l1,l2k1,k2,l1l2tg=k2-k11+k1k2.,k1=k2,l1l2;k1k2=-1,l1l2.21(1994)PQPQ(-1,1)(2,2),l:x+my+m=0PQ,m.1:m,m.x+my+m=0y=-1mx-1(m0),m.,PQy=13x+43,k=13.y=-1mx-1A(0,-1),lPQ11(1):kPQklkAQ13-1m23.m-3m-23.2:lPQ,lPQm.,lPQ,lPQR(-7mm+3,4-mm+3).RPQ:-7mm+32,4-mm+32,-3m-23,m.:,m-3,lPQ.3:RPQ,RPQ,PRRQ=,-1,R(-1+21+,1+21+).Rl-1+21++m1+21++m=0,=1-2m2+3m(m-23).-1m.,,.1,.;2,m,;3lPQ,RPQ,m.2(1994),|x+y|2a+|x-y|2b=1(a,b9320008©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.)()(A).(B).(C).(D).|x+y|2a+|x-y|2b=1(a,b)():l1:x+y2a+x-y2b=1(x+y0,x-y0),(a+b)x-(a-b)y=2ab(x+y0,x-y0);l2:x+y2a-x-y2b=1(x+y0,x-y0),-(a-b)x+(a+b)y=2ab(x+y0,x-y0);l3:-x+y2a-x-y2b=1(x+y0,x-y0),-(a+b)x+(a-b)y=2ab(x+y0,x-y0);l4:-x+y2a+x-y2b=1(x+y0,x-y0),(a-b)x-(a+b)y=2ab(x+y0,x-y0).,,,,.,l1l3,l2l4.l1,l2l1l2A(a,a),l1l4B(b,-b),l3l4C(-a,-a)l3l2D(-b,b).AC,BDO,AO=CO,BO=DO,ACBD.ab,ACBD,.3(1999)ax+by+c=0a,b,c{-3,-2,-1,0,1,2,3}3,,?:k0,-ab0.,a0,b0.c=0c0.1)c=0,a1,2,3,b-1,-2,-3,3x-3y=0,2x-2y=0x-y=0,33-2=7;2)c0,a1,2,3,b-1,-2,-3,ca,b,334=36.1),2)43.4(1999),,(|x|-1)2+(|y|-1)22(x,y)()(A)16.(B)17.(C)18.(D)25.(|x|-1)2+(|y|-1)22(|x|-1,|y|-1)(0,0),(0,1),(1,0),(0,-1)(-1,0),(x,y)16,(A).5ABCA(1,2),B(2,3),C(3,1),l,ABCl.A,B,Cld,1)lx,lx=0,d=14;2)lx,lk,lkx-y=0,d=14k2-22k+141+k2,(d-14)k2+22k+(d-14)=0.kR,=222-4(d-14)20,3d25,dmax=25(k=-1).1),2)dmax=25.6(1999)A(1,2),(5,-2)y2=4xB,C,:ABC.:,BAC=90.:kABkAC=-1.B,Cy2=4x,A,B(t2,2t),C(s2,2s),st,s1,t1,B,C,(5,-2),2t+2t2-5=2s+2s2-5,(1+t)(1+s)=-4,kABkAC=2-2t1-t22-2s1-s2=4(1+t)(1+s)=-1.BAC=90,ABC.,,.6B,C,,.1(3a+2)x-(a-1)y-1=0,a()(A)a1.(B)a0a1.(C)a1.(D)-1a2.2(1999)P(1,1)2.3(1995),y3xy13xx+y100.4(1994)A={(x,y)|(x-3)2+(y-4)2(52)2},B={(x,0420008©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.(,528454):O12-4:A:0488-7395(2000)08-0041-01:1999-11-15:(1963),,,,.,,,,.,.1()xR,f(x)=x2+3x+3+x2-3x+3.f(x)=(x+32)2+(0+32)2+(x-32)2+(0-32)2.P(x,0),A(-32,-32),B(32,32),|PA|+|PB|.12|PA|+|PB|=|AB|=23.2(1990)f(x)=x4-3x2-6x+13-x4-x2+1.f(x)=(x4-4x2+4)+(x2-6x+9)-(x4-2x2+1)+x2=(x2-2)2+(x-3)2-(x2-1)2+x2y=x2,y=x2P(x,x2)A(3,2)B(0,1).||PA|-|PB|||AB|=10,23f(x)10.3()x,yu(x,y)=x2+81x2-2xy+18x2-y2.u(x,y)=x2-2xy+y2+[(9x)2+29x2-y2+(2-y2)]-2=(x-y)2+(9x+2-y2)2-2.P1(x,9x),P2(y,-2-y2),xR,x0,P1;yR,|y|2,P2,2,|P1P2|32-2,(32-2)2-2=6.y)|(x-4)2+(y-5)2(52)2},AB.5ABCDABy=2x-17,C,Dy=x2,.64,y=x,y=2x,.7ABCC(x0,y0)x2+y28+2y,y3.ABx,Q(0,1)AC,BC1,ABC.1(C).23.32551.47.5801280.6A(a,a),B(b,2b),P(x,y),a=2(2x-y),b=2(y-x),|AB|=4]25x2-36xy+13y2=4.7A(a,0),AC-y0(x-a)+y(x0-a)=0,QAC1,(y20-2y0)a2+2x0y0a-y20=0,a1,a2A,B.SABC=12y0|AB|=y022x20+y20-2y0y0-28y0y0-2=8(1+2y0-2)8(1+23-2)=62(y03,C(5,3)).1420008©1994-2007ChinaAcademicJournalElectronicPublishingHouse.Allrightsreserved.