拋物線的焦點(diǎn)弦性質(zhì)及其證明過程

1、有關(guān)拋物線焦點(diǎn)弦問題的探討AB82過拋物線y2px(p>o)的焦點(diǎn)F作一條直線L和此拋物線相交于A(xi,y。、B(x2,y?)兩點(diǎn)結(jié)論1：ABx1x2pABAF結(jié)論2:若直線BF(xi)(x2)22L的傾斜角為，則弦長ABxix22P.2sin證：(1)若-時，直線L的斜率不存在，此時AB為拋物線的通徑AB2p結(jié)論得證(2)若一時,設(shè)直線L的方程為：y2P、一(x)tan即xycot-代入拋物線方程得2p2,yiy22pcot由弦長公式得ABVicot2yiy22p(icot2)2P.2sin結(jié)論3:過焦點(diǎn)的弦中通徑長最小sin2i2p2pAB的最小值為2p,即過焦點(diǎn)的弦長中通徑長最短

2、sin23i-一 OF AF sin2結(jié)論4:謂力為定值)S OAB S OBFci.Soaf一OFBFsin2i-,OFAFBFsin2S2p3SOABpi-OOFABsin2工衛(wèi)2P22sin2sin2P2sin結(jié)論5：(i)y1y2(2)2Y-PXiX242證Xi”2P,X22y2而X1X2(yiy2)24P2P2結(jié)論6:以AB為直徑的圓與拋物線的準(zhǔn)線相切證：設(shè)M為AB的中點(diǎn)，過A點(diǎn)作準(zhǔn)線的垂線AAi,過B點(diǎn)作準(zhǔn)線的垂線BBi,結(jié)論結(jié)論結(jié)論過M點(diǎn)作準(zhǔn)線的垂線MMi,由梯形的中位線性質(zhì)和拋物線的定義知MMi7：連接AiF、AAi同理8：(i)(4)(5)AAiBBiAFBFAB故結(jié)論得證B

3、iF則AiFAF,AAiFBiFAFAiAAi/OFAAiFAiFOAiFOAiFABiFOBiFBAiFBi90AiFBiFAMiBMi(2)MiFAB|MiFAFBF設(shè)AMi與AiF相交于H,MiB與FBi相交于Q則Mi,Q,FAMiMiB24MiM證：由Z論（6）知Mi在以AB為直徑的圓上AMiBMiAiFBi為直角三角形，Mi是斜邊AiBi的中點(diǎn)H四點(diǎn)共圓9:證:AiMiAAiFMiFMiFAiFBiAFMiFFAiMiABAFBF90BF(i)A、O、Bi(3)(4)設(shè)直線AO因?yàn)镸iFAiAAiMi所以AAiM1AlFAFAFAi90AFAiAiFMi90AMiBMi三點(diǎn)共線AMi

4、B90又AiFBiFMi,Q,F,H四點(diǎn)共圓,BBi2MMi(2)B,O與拋物線的準(zhǔn)線的交點(diǎn)為設(shè)直線BO與拋物線的準(zhǔn)線的交點(diǎn)為koAyiXiyi2yi2p2Pyi,koBiy2p2AMi4MMiAi三點(diǎn)共線Bi,則BBi平行于Ai,則AAi平行于所以koA2P2PV2MiB2%81所以三點(diǎn)共線。同理可征（2）（3）AB(4)_i_FAFBi 2結(jié)論i0：證：過A點(diǎn)作AR垂直X軸于點(diǎn)R,過B點(diǎn)作BS垂直X軸于點(diǎn)S,設(shè)準(zhǔn)線與x軸交點(diǎn)為E,因?yàn)橹本€L的傾斜角為則EREFFRPAFcos同理可得1cosBF結(jié)論11：(1)線段EF平分角AFAF1cos112FA-FB-pAFPEQBF(4)當(dāng)一時AE

5、BE,當(dāng)2證：BB"/EFAAB1EBFEAFAAA1EBB1E9011cosAF|PAE(3)KaeKbe0BE一時AE不垂直于BE2BFB1B,FAA1AB1EEAB1BA1aA1EA相似于B1EBA1EA=B1EBAEF+A1EA=BEF+B1EB=90AEF=BEF即EF平分角PEQAFBFAEBE直線AE和直線BE關(guān)于X軸對稱Kae+Kbe=0(4)當(dāng)一時，AF=EF=FBAEB=902時，設(shè)直線邙方程為ykx-p2將其代入方程y2px彳#k2x2-p(k22)x,22T0設(shè)A(X1，y1)，B(X2，y2)則x1X2Pk22PXiX2二一4假設(shè)AEBE則KaeKbe=TX

6、iy1衛(wèi)2X2y2_p2即y1y2X1-2X2X2XiX2k21x1x22x1x2kk22k212k220不可能假設(shè)錯誤結(jié)論得證結(jié)論12:過拋物線的焦點(diǎn)作兩條互相垂直的弦AB、CD,|CD|12p推廣與深化:深化1:性質(zhì)5中，把弦AB過焦點(diǎn)改為AB過對稱軸上一點(diǎn)E(a,0),則有y1y22Pa.22證：設(shè)AB方程為my=x-a,代入N. |AB|Px.得：y2Pmy2ap0,.丫422Pa.深化2:性質(zhì)12中的條件改為焦點(diǎn)弦AB不垂直于|FR|1x軸，AB的中垂線交x軸于點(diǎn)R,則1ABi2證明：設(shè)ABy的傾斜角為a,直線AB的方程為：tga(xp)2,2,22Px得：tga(xPxx即：2 、

7、x(p 2pctg a)由性質(zhì)|AB|xx2P2p2pctg2P2sina,又設(shè)AB的中點(diǎn)為M,則|FM|xx2pI2Icosa,2Pctgacosa|FE|P.2- sin a,2深化3:過拋物線的焦點(diǎn)F作n條弦AiBA2B2、AnBn,且它們等分周角2兀，則有(1)i1|AiFI|FBi1為定值;(2)n1i1|AiBi|為定值.證明：(1)設(shè)拋物線方程為1PcosA1Fx由題意A2FX一，A3FXnAnFxa1cosa所以IA1FIIFB1Icos(Pa)21cosa2.2sina2P，.2,sin (a.2/n1sin(a)同理IA2FIIFB2I'|AnF|FBn|2P易知2_sina22sin(a-)sin(an2一)n.2sin(a2sin2(a-)1sinani|AiF|FBi|-PP22n1sin2(a)n2P2p2|AiBi|(2)P1cosaP2P2P1cos(a)1cos2asin2a21 s