极点极线专题 (学生版)

=1m+=1m+a2k2(x2+2kma2x+a2(m2-b2(=0(y=kx+则x1+x2=-,x1x2=y1+y2=kx1+m+kx2+m=k(x1+x2(+2m=-+=;=(kx1+m(⋅(kx2+m(=k2x1x2+km(x1+x2(+m2=-+x1y2+x2y1=x1(kx2+m(+x2(kx1+m(=2kx1x2+m(x1+x2(=-=-;=1+k2|x1-x2|=1+k2⋅、(x1+x2(2-4x1x2=1第1页(x=ty+m2+b2t22+b2t2则y1+y2=-2tmb2y12+b2t22+b2t2x1+x2=ty1+m+ty2+m=t(y1+y2(+2m=-+=;=(ty1+m(⋅(ty2+m(=t2y1y2+tm(y1+y2(+m2=-+ x1y2+x2y1=y2(ty1+m(+y1(ty2+m(=2ty1y2+m(y1+y2(=-=-;Δ=4t2m2b4-4b2(m2-=1+t2|y1-y2|=1+t2⋅、(y1+y2(2-4y1y2=1+t2⋅(A==1m2-=1m-a2k2(x2--a2k2(x2-2kma2x-a2(m2+b2(=0则x1+x2=-,x1x2=-y1+y2=kx1+m+kx2+m=k(x1+x2(+2m=-+=-;=(kx1+m(⋅(kx2+m(=k2x1x2+km(x1+x2(+m2=--+-+ x1y2+x2y1=x1(kx2+m(+x2(kx1+m(=2kx1x2+m(x1+x2(=-+-=--;第2页2-y22=1t2-a2(y2+2tmb2y+b2(m2-a2(=2-y22t2-a2则y1+y2=-2tmb2y12t2-a2x1+x2=ty1+m+ty2+m=t(y1+y2(+2m=-+=--;=(ty1+m(⋅(ty2+m(=t2y1y2+tm(y1+y2(+m2=---+ =-b2t2-a2;x1y2+x2y1=y2(ty1+m(+y1(ty2+m(=Δ=4t2m2b4-4b2(m2-a=1+t2|y1-y2|=1+t2⋅、(y1+y2(2-4y1y2=1+t2⋅(A=b+2(km-p(x+m2=0则x1+x2=-,x1x2=y1+y2=kx1+m+kx2+m=k(x1+x2(+2m=-+=;=(kx1+m(⋅(kx2+m(=k2x1x2+km(x1+x2(+m2=m2-+m2=;x1y2+x2y1=x1(kx2+m(+x2(kx1+m(=2kx1x2+m(x1+x2(=-=;Δ=4(km-p(2-4k2m2=4p(p-2km(;-2tpy-2pm=0则y1+y2=2tp,y1y2=-2pmx1+x2=ty1+m+ty2+m=t(y1+y2(+2m=2pt2+2m;第3页2=(ty1+m(⋅(ty2+m(=t2y1y2+tm(y1+y2(+m2=-2pmt2+2pmt2+m2=m2;x1y2+x2y1=y2(ty1+m(+y1(ty2+m(=2ty1y2+m(y1+y2(=-4pmt+2pmt=-2pmt;Δ=4t2p2-4(-2pm(=4p(pt2+2m(;线的斜率为k=-【证明】：设切线方程为y=k(x-x0(+y0即y=kx+m(m=y0-kx0)∵点P(x0,y0(为椭圆=1(a>b>0(上一点=1，则x-a2=--b2=-则Δ=4k2m2a4-4a2(m2-b2((b2+a2k2(=4a2b2(b2+a2k2-m2(=0,即b2+a2k2-(y0-kx0(2=0整理得：(x-a2(k2-2x0y0k+y-b2=0,即yk2+2x0y0k+x=0,线方程为+=1.第4页设切线方程为y-y0=k(x-x0(,又k=-,则y-y0=-(x-x0(2y0y-a2y=b2x-b2x0x,b2x0x+a2y0y=b2x+a2y=a2b2则P(x0,y0(处的切线方程为③椭圆方程为+=1(a>b>0(,点P(x0,y0(为椭圆外一点，过点P(x0,y0(作椭圆的切线PA∴切线PA和切线PB方程分别为∵点P(x0,y0(在直线PA和PB上故切点弦AB的方程为A(x1点为点M，则点M的轨迹方程为+=1.第5页又点P(x0,y0(在直线AB上即点M满足方程故点M的轨迹方程为【证明】：设切线方程为y=k(x-x0(+y0即y=kx+m(m=y0-kx0)∵点P(x0,y0(为双曲线=1(a>0,b>0(上一点则Δ=4k2m2a4+4a2(m2+b2((b2-a2k2(=4a2b2(b2-a2k2+m2(=0,即b2-a2k2+(y0-kx0(2=0整理得：(x-a2(k2-2x0y0k+y+b2=0,即yk2-2x0y0k+x=0,第6页2x-a2y=a2b2设切线方程为y-y0=k(x-x0(,又k=,则y-y0=(x-x0(2y0y-a2y=b2x0x-b2x,b2x0x-a2y0y=b2x-a2y=a2b2则P(x0,y0(处的切线方程为∴切线PA和切线PB方程分别为∵点P(x0,y0(在直线PA和PB上故切点弦AB的方程为A(x1第7页又点P(x0,y0(在直线AB上即点M满足方程故点M的轨迹方程为①抛物线方程为y2=2px(p>0(,点P(x0,y0(为抛物线上一点，过点P(x0,y0(作抛物线的切线，则切【证明】：设切线方程为y=k(x-x0(+y0即y=kx+m(m=y0-kx0)∵点P(x0,y0(为抛物线y2=2px(p>0(上一点联立>0(,消元得：k2x2+2(km-p(x+m2=0则Δ=4(km-p(2-4k2m2=4p(p-2km(=0,即p=2k(y0-kx0(0k2-2y0k+p=0,即k2-2y0k+p=0②抛物线方程为y2=2px(p>0(,点P(x0,y0(为抛物线上一点，过点P(x0,y0(作抛物线的切线，则切线方程为y0y=p(x0+x(.【证明】：∵点P(x0,y0(为抛物线y2=2px(p>0(上一点设切线方程为y-y0=k(x-x0(,又k=,则y-y0=(x-x0(第8页∴y0y-y=p(x-x0(,即y0y=p(x-x0(+y=p(x-x0(+2px0=p(x0+x(.则P(x0,y0(处的切线方程为y0y=p(x0+x(.③抛物线方程为y2=2px(p>0(,点P(x0,y0(为抛物线外一点，过点P(x0,y0(作抛物线的切线PA和∴切线PA和切线PB方程分别为y1y=p(x1+x(、y2y=p(x2+x(∵点P(x0,y0(在直线PA和PB上1y0=y2y0=p(x1+x1y0=y2y0=即A、B两点都满足方程y0y=p(x0+x(故切点弦AB的方程为y0y=p(x0+x(④抛物线方程为y2=2px(p>0(,点P(x0,y0(为抛物线内一点，过点P(x0,y0(作抛物线的一条弦AB，与抛物线分别交于点A(x1,y1(、B(x2,y2(，过点A(x1,y1(和点B(x2,y2(分别作抛物线的切线，切线交点为点M，则点M的轨迹方程为y0y=p(x0+x(.∴直线AB的方程为yMy=p(xM+x(又点P(x0,y0(在直线AB上∴yMy0=p(xM+x0(即点M满足方程y0y=p(x0+x(故点M的轨迹方程为y0y=p(x0+x(第9页若交比为-1,则称为调和比.交比为-1的线束称为调和线束,点列称为调和点列.一般地,若外分点.==λ,则称顺序点列A,C,B,D四点构成调和点列，其中A,B叫做基点,C,D叫做内、外分点.(调和关系=+.则顺序点列A,C,B,D四点仍构成调和点列。所以称A,B和C,D称为调和共轭.①调和性:+=∴+= 1+1=2也成立.∴-1=1-即∴-1=1-即+=2∴+=⑴==λ+|MB|-|MC||MD|-|MB|:==-交比：如图,过点O的四条直线被任意直线l所截的有向线段之比称为线束OA、OC、OB、OD或点列A,C,B,D的交比.【证明】:令线束O(a,b,c,d(分别交l于A,B,C,D,:/=-/=-●【证明】:令线束O(a,b,c,d(分别交l于A,B,C,D,交l/于A/,B/,C/,D/.:EF=BF,即点F为BE的中点:CF=EF,即点F为CE的中点:AE=EF,即点E为AF的中点:DE=EF,即点E为DF的中点线.2(在椭圆=1(a>b>0(上可得2x+y2+=λ2⋯②①-②得：+=(1+λ(⋅(1-λ(Q=x=xB=xA即xP⋅xQ-x+xP⋅xA-xQ⋅xA=xP⋅xA+x-xP⋅xQ-xQ⋅xAPQ=x=x是直线AB(即切点弦所在的直线),点P点N，则点M、N必定在点P对应的极线上.线EF(即切点弦所在的直线)，则点M,N一定在点P对应的极线EF上,此时称点P为极点，直线MN直线TS(即切点弦所在的直线).则点P,N一定在点M对应的极线TS上,此时称点M为极点，直线PM,此时称点N为极点，直线PM称为极点N对应的极线.我们把△PMN称为“自极三角形”.自极三角形△PMN：极点P一一极线MN；极点M一一极线PN;极点N一一极线PM.①曲线上的点A(B、M、N(对应的极线就是切线PA;①AB和CD的交点P落在无穷远处；2+Cy2+2Dx+2Ey+F=0,则称点P(x0,y0)和直线l:Ax0x+Cy0y+D(x+x0)+E(y+y0)+F=0是圆锥曲线Γ的一对极点和极线.0极线方程.⑶对于抛物线y2=2px，与点P(x0,y0)对应的极线方程为y0y=p(x0+x).当P(x0,y0)为其焦点0y=p(x0+x)变为x=−,恰为抛物线的准线。对于任意的二次曲线C：Ax2+By2+Cxy+Dx+Ey+F=0,平面内任一点P(x0,y0)对关于曲线C对Ax0x+By0y+C⋅+D⋅+E⋅+F=0x2→x0x0y0【证明】：设圆锥曲线Γ的方程为Ax2+Cy2+2Dx+2Ey+F=0两边同时对x求导得2Ax+2Cyy/+2D+2Ey/=0，解得y/=-，于是曲线Γ在P点处的切线斜率k=-，故切线l的方程为y-y0=-(x-x0)化简得，Ax0x+Cy0y-Ax-Cy+Dx+Ey-Dx0-Ey0=0⋯①∴Ax+Cy+2Dx0+2Ey0+F=0，即Ax+Cy=-2Dx0-2Ey0-F⋯②由①②可得曲线Γ在P点处的切线l的方程为Ax0x+Cy0y+D(x0+x)+E(y0+y)+F=0.0由①可知，点M处的切线方程为:Ax1x+Cy1y+D(x1+x)+E(y1+y)+F=0点N处的切线方程为Ax2x+Cy2y+D(x2+x)+E(y2+y)+F=0∵点P在切线PM上∴Ax0x1+Cy0y1+D(x0+x1)+E(y0+y1)+F=0∵点P在切线PN上∴Ax0x2+Cy0y2+D(x0+x2)+E(y0+y2)+F=0即点M(x1,y1),N(x2,y2)都在直线Ax0x+Cy0y+D(x0+x)+E(y0+y)+F=0上故切点弦MN所在的直线方程为Ax0x+Cy0y+D(x0+x)+E(y0+y)+F=0.设S(x1,y1),T(x2,y2)，M(m,n)如图所示：由①可知，曲线在点S(x1,y1)处的切线方程为Ax1x+Cy1y+D(x1+x)+E(y1+y)+F=0曲线在点T(x2,y2)处的切线方程为Ax2x+Cy2y+D(x2+x)+E(y2+y)+F=0，∵点M(m,n)在切线SM上∴Ax1m+Cy1n+D(x1+m)+E(y1+n)+F=0∵点M(m,n)在切线ST上∴Ax2m+Cy2n+D(x2+m)+E(y2+n)+F=0即点S(x1,y1),T(x2,y2)都在直线Axm+Cyn+D(x+m)+E(y+n)+F=0上故直线ST的方程为Axm+Cyn+D(x+m)+E(y+n)+F=0.又直线ST过点P(x0,y0)∴Ax0m+Cy0n+D(x0+m)+E(y0+n)+F=0即点M(m,n)在直线Ax0m+Cy0n+D(x0+m)+E(y0+n)+F=0∴两切线的交点的轨迹方程式Ax0x+Cy0y+D(x0+x)+E(y0+y)+F=0.点P对应的极线.设点P(t,n(,则直线MN的方程为∵点P(m,t(,∴直线MN的方程为又点MM(n,0(在直线 =1,即mn=a22+ny2=1(m>0,n>0且m≠n)，设点P(x0,y0(，则点P对应的极线方程为l:mx0x+ny0y=1.设M(x0,t(,E(xE,yE(,A(x1,y1(,B(x2,y2(,直线lAB:y=kx+b(k≠0(,则有：y0=kx0+b-y-0-(t(mx0∴x1+x2=-,x1x2=【证明】：设点P(n,t(，则点P对应的极线方程为=1.设M(n,s(,E(xE,yE(,A(x1,y1(,B(x2,y2(,直线lAB:y=kx+m(k≠0(,则有：t=kn+m联立1得：(b2-a2k2(x2-2kma2x-a2(b2+m2(=0y1+y2=k(x1+x2(+2m=--x1y2+x2y1=x1(kx2+m(+x2(kx1+m(=2kx1x2+m(x1+x2(=-+-=--【证明】：设点P(n,t(，则点P对应的极线方程为l:ty=p(n+x(.设M(n,s(,E(xE,yE(,A(x1,y1(,B(x2,y2(,直线lAB:x=my+k,则有：n=mt+k-2pmy-2pk=0∴y1+y2=2pm,y1y2=-2pkx1y2+x2y1=(my1+k(y2+(my2+k(y1=2my1y2+k(y1+y2(=-2pmkx1y2+x2y1-sx1y2+x2y1-s(x1+x2(-n(y1+y2(+2sn-2pmk-s(2pm2+2k(-2pmn+2sn==x1x2-n(x1+x2(+n2k2-n(2pm2+2k(+n2E(m,0(，(点E不在椭圆端点和椭圆中心(若M是E点对应极线上任一点，则kMA,kME,kMB成MA+kMB=2kMEMA+kMB=2kMP=0极线x=；椭圆在点A处的切线与极线x=交于点N，过点N作直线AP的垂线MN，垂足为切线AN与极线的交点由P(t,0(,MN⏊PA可得直线MN的斜率为kMN=-极线x=.双曲线在点A处的切线与极线x=交于点N，过点N作直线AP的垂线MN，垂足切线AN与极线的交点由P(t,0(,MN⏊PA可得直线MN的斜率为kMN=-t；抛物线在点A处的切线与极线x=-t交于点N，过点N作直线AP的垂线MN，垂足为M，则直,y1(，则切线AN的方程为y1y=p(x1+x(切线AN与极线x=-t的交点N(-t,由P(t,0(,MN⏊PA可得直线MN的斜率为kMN=∴直线MN的方程为(x+t(∴直线MN的方程过x轴上一定点Q(p-t,0(当x1=t时，直线MN的方程为y=0,也过点Q(p-t,0(∴直线MN恒过x轴上一定点Q(p-t,0(MN为直径的圆必过焦点F.如图所示：,N∵直线FN为点M对应的极线又点N在直线FN上=1,即y1y2=-∴kFN⋅kFM==-1,即FM⏊FN∴以线段MN为直径的圆必过焦点F,N∵直线FN为点M对应的极线又点N在直线FN上∴以线段MN为直径的圆的方程为(x-2+(y-y1((y-y2(=0x-2=⋅(a2-t2(,即x-=±、a2-t2,即∴AP⋅PB+PE⋅PB=AP⋅PB-PE⋅AP,即PB=-AP∵PB,AP均为长度∴PB=-AP不可能成立，即假设不成立故P,Q,R三点一定共线要证点Q在直线只需要证AC与直线l的交点Q