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杨敬民(,610041):2009-03-16:2009-09-11(本讲适合高中),.,(),..1=ep1-ecos,,e,pF.0e1,FFx;e=1,FFx;e1,FFx,0.22.1定值问题1O,F,PQF.|OF|=a,|PQ|=b.OPQ.(1991,):1,FFx.=2a1-cos.P(1,)(0).1Q(2,+).|PQ|=1+2=2a1-cos+2a1-cos(+)=4asin2,4asin2=b.,sin=2ab.SOPF=12a|FP|sin,SOQF=12a|FQ|sin,SOPQ=SOPF+SOQF=12a(|FP|+|FQ|)sin=12absin=12ab2ab=aab.2Pi(i=1,2,!,n)n,F,FP1,FP2,!,FPnFn.:ni=11|FPi|=nep.:FFx.P1(1,).P22,+2n,P33,+4n,!,Pnn,+2(n-1)n.,|FP1|=1=ep1-ecos.6,|FP2|=2=ep1-ecos+2n,|FP3|=3=ep1-ecos+4n,!!|FPn|=n=ep1-ecos+2(n-1)n.ni=11|FPi|=ni=11-ecos+2(i-1)nep=1epn-eni=1cos+2(i-1)n,,ni=1cos+2(i-1)n=0(,).,ni=11|FPi|=nep.2.2范围问题3C:y2=4x,FC,FlCAB,FB=AF.[4,9],ly.:l,FB=AF,[4,9].FBAF,,FBAF.,.22,FFx.,BFx.=21-cos.,B.(1)B,B(1,)BFx=,02.A(2,+).,|FB|=1=21-cos,|AF|=2=21+cos.FB=AF,|FB|=|AF|.=|FB||AF|=1+cos1-cos.[4,9],,4∀1+cos1-cos∀902.cos35,45.lkk2=tan2=1cos2-1916,169.02,34∀k∀43.(2)B,-43∀k∀-34.ly=k(x-1).lyb=-k-43,-34#34,43.2.3最值问题343,Fy2=4x,ABO,FAFB=0.AFBFCD.ABCD.(2008,)7200911:3,FFx.=21-cos.A(1,)(0).B2,+2C(3,+)D4,+32.|AC|=1+3=21-cos+21-cos(+)=4sin2.,|BD|=4cos2.SABCD=12|AC||BD|=124sin24cos2=32sin22.sin2=1,=4,SABCD32.注:,,.2.4坐标系的互化5x2-y2=1FMN,PMN.(1)P;(2)PPQMNxQ,:|MN||FQ|=2.:(1)MNF,,.F(2,0)Fx.=11-2cos(R).M(1,)N(2,+),-2,-4#4,2,P(,).1=11-2cos,2=11+2cos.,=1-22=2cos1-2cos2.MNP..-2cos2=2cos2-22cos2=2cos2sin2-2cos2=2cos0y2-(x-2)2=2(x-2).x-222-y2=12(x∃2).(2)|MN|=1+2=11-2cos+11+2cos=21-2cos2.|FQ|=|FP|cos=2cos1-2cos2cos=21-2cos2.,|MN||FQ|=22=2.注:.6C0:x2+y2=1C1:x2a2+y2b2=1(ab0).:ab,C1P,P,C0C1?.(2000,):,P,ab,.,,8xa+yb=1bx+ay=ab.C0aba2+b2=1a2+b2=a2b21a2+1b2=1.,ab1a2+1b2=1.:必要性.44,PQRSC0,,PQRS.,OPOQ,OPQ1.,|OP||OQ||OP|2+|OQ|2=1.%OOx.P(1cos,1sin).Q2cos+2,2sin+2,Q(-2sin,2cos).PQ,21cos2a2+21sin2b2=1,22sin2a2+22cos2b2=1,cos2a2+sin2b2=121,sin2a2+cos2b2=122.121+122=1a2+1b2.%1221+22=11121+122=111a2+1b2=11a2+1b2=1.充分性.1a2+1b2=1,PC1.POC1PR,OPRQS.PQRSC1.OOx.P(1cos,1sin)Q(-2sin,2cos)21cos2a2+21sin2b2=1,22sin2a2+22cos2b2=1.121+122=1a2+1b2,1|OP|2+1|OQ|2=1.RtPOQ,OPQh.h=1.,OPSQRRS1.PQRSC0.,ab1a2+1b2=1.注:,,1a2+1b2=1.,.练习题1.|A1A2|=6,|F1F2|=42,F1MN.F2F1M=(0∀).|MN|,=.(提示:=13-22cos.M(1,)N(2,+).|MN|=13-22cos+13-22cos(+)9200911=69-8cos2.,69-8cos2=2.cos=&32.0∀,=656.)2.C:x2a2-y2b2=1(a0,b0)F,F3CAB.AF=4FB,C().(A)65(B)75(C)58(D)95(提示:=ep1-ecos.1=|AF|=ep1-ecos3,2=|FB|=ep1-ecos3+.AF=4FB,|AF||FB|=4,12=1-ecos3+1-ecos3=4.e=65.(A).)3.x2-y22=1FlAB.|AB|=3,.(提示:=21-3cos.AB.|AB|=1+2=21-3cos+21-3cos(+)=41-3cos2∃4(=2,),,x,4.3,(1),,,,;(2),,,x,,|AB|==4.=4,4,=4.)4.PQMNx22+y2=1,Fx.PFPQ,MFFN,PFMF=0.PMQN.(提示:4.169∀SPMQN∀2.)5.x224+y216=1,l:x12+y8=1,Pl,OPR,QOP,|OQ||OP|=|OR|2.Pl,Q,.(提示:l=242cos+3sin,2=482+sin2.Q(,)R(1,)P(2,).=|OQ|∋02=21.Q(2+sin2)-2(2cos+3sin)=0(∋0).(x-1)252+(y-1)253=1.,(),(1,1),xy,102,153.)10
本文标题:非圆二次曲线极坐标方程的应用
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