构造齐次方程巧解一类圆锥曲线问题

1010(530021),.,,.:C:f(x,y)=Ax2+Bxy+Cy2+Dx+Ey+F=0,L:lx+my+n=0.LCP,QPQ:(A+C)n2-(Dl+Em)m+F(l2+m2)=0(*).:CP,Q,F=0,P(-nl,0),Q(0,-nm)f(x,y)=0,(*).P,Q,L.n0,lx+my+n=0,1=lx+my-n.f(x,y)=0Ax2+Bxy+Cy2+(Dx+Ey)(lx+my-n)+F(lx+my-n)2=0.!()∀!,LC(x0,y0),Ax20+Bx0y0+Cy20+(Dx0+Ey0)#(-lx0+my0n)+F(-lx0+my0n)2=Ax20+Bx0y0+Cy20+(Dx0+Ey0)+F=0.(x0,y0)!,!LC(x0,y0),!:(A-Dln+Fl2n2)x2+(B-Dmn-Eln+2Flmn2)xy+(C-Emn+Fm2n2)y2=0.:y=x,y=x,=0,C-Emn+Fm2n2A-Dln+Fl2n2=-1(A+C)n2-(Dl+Em)n+F(l2+m2)=0.LCA,B,,x,y,xy(),AB..∃1%ABy2=4px(p0),OA&OB,OM&AB,M,.:AB:mx+ny=1,y2=4pxy2=4px(mx+ny)4pmx2+4pnxy-y2=0.∀OA&OB,k1k2=-4pm=-1.m=14p!OMy=nmx,y=4pnx∋!∋mx+ny=0x2+y2=4px(x-2p)2+y2=4p2(x0).M(2p,0),2p().∃2%Rx,(ABC,,BC4x+1011y-20=0.())R;(∗)M,MlRPQ,+POQ?.:())y2=2px(p0),A,B,C(x1,y1),(x1,y2),(x3,y3),y2=2px,y=-4x+20.8x2-(p+80)x+200=0.x2+x3=(p+80)/8,y2+y3=-4,(x2+x3)+40=-p/2,x1=(11p-80)/8,y1=p/2,y2=2px,p=8.Ry2=16x.(∗)M(x0,y0),PQlx+my-lx0=0y2=16xy2=16x(lx+mylx0+my0)16lx2-16mxy-(lx0+my0)y2=0.-(lx0+my0)16l=-1l(x0-16)+my0=0.(16,0).∃3%y2=p(x+1)(p0),x+y=mx.()):;(∗)QR,OQ&OR,pmf(m).:())x=-1-p4,x+y=mx(m,0),m-1-p44m+p+40,y2=p(x+1),x+y=m.x2-(2m+p)x+(m2-p)=0.=(2m+p)2-4(m2-p)=p(4m+p+4)0.(∗)l:x+y=my2=p(x+1)y2=pxx+ym+p(x+ym)2.(mp+p)x2+(mp+2p)xy+(p-m2)y2=0,p-m2mp+p=-1,p=m2m+2=f(m).∃4%C0:x2+y2=1C1:b2x2+a2y2=a2b2(ab0),:ab,C1P,P,C0,C1?.:x2+y2=1(x1,y1),PQx1x+y1y=1,PQO(,,),PQb2x2+a2y2=a2b2b2x2+a2y2=a2b2(x1x+y1y)2(b2-a2b2x21)x2-2a2b2x1y1xy+(a2-a2b2y21)y2=0,a2-a2b2y21b2-a2b2x21=-1,a2+b2=a2b2#(x21+y21)=a2b2.a,b1a2+1b2=1.∃5%,x,3/5P,Q.OP&OQ,|PQ|=4,.:b2x2-a2y2=a2b2!ly=35x-35c,(c2=a2+b2).3x-5y-3c=0∋∋!b2x2-a2y2=a2b2(3x-5y3c)(3c2b2-3a2b2)x2+215a2b2xy-(3c2a2+5a2b2)y2=0-(3c2a2+5a2b2)3c2b2-3a2b2=-13a4+8a2b2-3b4=0(a2+3b2)(3a2-b2)=0.b2=3a2,y=35x-35#2a3x2-y2=3a212x2+12ax-27a2=0.PQM(x0,y0),x0=12(x1+x2)=-a/2,y0=-35#52a.|OM|2=a24+35#254a2=4a2=4.1012a2=1.3x2-y2=3.∃6%A(1,2)(5,-2)y2=4xBC,(ABC().A.B.C.D.:,A,(y,+2)2=4(x,+1),y,2+4y,-4x,=0!(5,-2)(4,-4),BCm(x,-4)+n(y,+4)=0.mx,+ny,=4m-4n!y,2=4(x,-y,)mx,+ny,4(m-n)mx,2+(-m+n)x,y,-my,2=0.kABkAC=-1.(ABC,C.∃7%lC:y2=4(x-1)A,B,ABCF,l.:ly=kx(k0),Cy2=4(x-1),yx!:k2x2-4x+4=0∋:k2y2-4dy+4k2=0.=16(1-k2)0-1k1.!∋A,B:k2(x2+y2)-4x-4ky+4k2+4=0−(2k2,2k)l:y=kx.−AB.CF(2,0),4k2-8+4k2+4=0k=.22/(-1,1)ly=.22x.∃8%l:y=x+bC:x2a2+y2a2-1=1(a1)AB,lCF2,CF1.AF1&BF1,C.:C1,F1(-1,0),F2(1,0),lF2,b=-1.ly=x-1,C,yx(2a2-1)x2-2a2x+2a2-a4-1=0(2a2-1)y2+2(a2-1)y+2a2-a4-1=0,AB:(2a2-1)(x2+y2)-2a2x+2(a2-1)y-2a4+4a2-1=0.(a22a2-1,1-a22a2-1)l:y=x-1,AF1&BF1,ABF1(-1,0),a4-4a2+1=0(a0)a2=2+3,Cx22+3+y21+3=1.∃9%C:x2+y2=34Q:x23+y2=1AB,A,B,O()r.:P(32cos!,32sin!)C,PQ,PC2xcos!+2ysin!=3QAB.2xcos!+2ysin!=3,x2+3y2=3.yx:4(sin2!+3cos2!)x2-123xcos!+9-12sin2!=04(sin2!+3cos2!)y2-43ysin!+3-12cos2!=0.AB(sin2!+3cos2!)(x2+y2)-33xcos!-3ysin!=0.O,r=1227cos2!+3sin2!(sin2!+3cos2!)2=32#1+8cos2!1+2cos2!,1013t=1+8cos2!/[1,3],r=23t2+3.t/[1,3],rt,t/[3,3],r,t=13,rmin=32;t=3,rmax=1,r32,1.∃10%(x+4)24-y212=1F,l,QFl,F450mQAB,ABO,,QO,,Qe.:,(-4,0),F(0,0),l:x=-3,Qabc,QG(c,0),a2c-c=3a2=c2+3c,b2=3c,Q(x-c)2c2+3c+y23c=1!,mF(0,0)450,m:y=x,!(c+6)x2-6cx-9c=0(c+6)y2-6cx-9c=0(0),A,B(c+6)(x2+y2)-6cx-18c=0,(3cc+6,3cc+6)m,ABO,G(c,0)O,,(c+6)c2+6c2-18c00c32,e2=c2a2=c23c+c2=1-33+c0e22-2,e(0,2-2).∃11%QO,x,35A,B,OA&OB|AB|=4,Q.:Qx2a2-y2b2=1!y=35(x-c)∋(a,b,c,c2=a2+b2)!∋(5b2-3a2)x2+6a2cx-3a2c2-5a2b2=0(5b2-3a2)y2+215b2cy+3b2c2-3a2b2=0.5b2-3a20(ba=35,lQ,Q,).A,B(5b2-3a2)(x2+y2)+6a2cx2+215b2cy-3a2c2+3b2c2-8a2b2=0.(-3a2c5b2-3a2,15b2c5b2-3a2)l.OA&OB,O,-3a2c2+3b2c2-8a2b2=03a4+8a2b2-3b4=0.(a2+3b2)(3a2-b2)=0.b2=3a2,c=2a.x2+y2+ax+15ay=0,|AB|=4,a2+15a2=4a2=1,b2=3.Q:x2-y23=1.:Ax+By+C=0(x-a)2+(y-b)2=r2PQ:Ax+By+c=0!(x-a)2+(y-b)2=r2∋.∃12%x2+y2-6y+m=0x+2y-3=0PQ,PQ,m.:x2+y2+x-6y+m=0x+2y-3=0x2+y2+2x-4y+m-3PO,(0,0),m=23..