高中數(shù)學(xué)圓錐曲線基礎(chǔ)知識點(diǎn)

高中數(shù)學(xué)圓錐曲線基礎(chǔ)知識點(diǎn)一、橢圓及其標(biāo)準(zhǔn)方程1、橢圓的定義:平面內(nèi)到兩定點(diǎn)F、F的距離之和等于常數(shù)(大于|FF|)的點(diǎn)的軌跡叫做橢1212圓(這兩個(gè)定點(diǎn)叫做橢圓的焦點(diǎn)，兩焦點(diǎn)的距離叫做焦距(兩種標(biāo)準(zhǔn)方程的比較22xy(1)表示焦點(diǎn)在x軸上的橢圓，焦點(diǎn)是F(-c,0),F(c,0);，,,,1(0)ab1222ab22yx(2)表示焦點(diǎn)在y軸上的橢圓，焦點(diǎn)是F(0,-c),F(0,c);，,,,1(0)ab1222ab在兩種標(biāo)準(zhǔn)方程中2221a,b,c的關(guān)系c=a-b不變，只須將(1)方程的x、y互換即可得到(2);?222?a,b，?可以根據(jù)分母的大小來判定焦點(diǎn)在哪一個(gè)坐標(biāo)軸上(?ce,離心率定義:橢圓的焦距與長軸長的比，叫做橢圓的離心率(離心率e的取值范圍:a0<e<1(方程圖形范圍，，對稱性關(guān)于軸、軸、坐標(biāo)原點(diǎn)對稱關(guān)于軸、軸、坐標(biāo)原點(diǎn)對稱A(-a,0)，A(a,0)(0,-a)，A(0,a)A1212頂點(diǎn)B(0,-b)，B(0,b)B(-b,0)，B(b,0)1212長軸長=2a,短軸長=2b離心率office,branchoffices(jurisdiction),riskmanagement,marketingmanagementsectorthroughsupervisionandinspectionfoundproblems,shouldbeassignedtheinvestigatorsarecorrectedinatimelymanner.27ththefifthchapterpenaltyunderanyofthefollowingacts,thentherelevantprovisionstopunishtheinvestigators,accordingtotheBank.Toconstituteacrimeshallbeinvestigatedforcriminalresponsibility:(A)onthebusinessthatarenotinvolvedintheinvestigation,issuedasurvey.(B)customercreditinformationarenotthoroughverification.(Iii)toparticipateincreditcustomersurveyisnotinplace,customersanddataisincomplete,untrue;heknowsbearacounterfeitedclientsissuingcredit.(D)doesnotprovideforduediligenceofcreditbusiness,pre-loaninvestigationform,concealingfactsorprovidingfalseinformationorshouldbefoundinanormalinvestigationfailedtodiscovertheriskfactors,leadtothereviewandapprovalpolicyerrors,loanrisk.(Five)onloanguaranteesofsurveynotinplace,notbyprovidesonarrived,andpledgerealforfieldverification,andassessment,andidentificationandregistration,notaccordingtoprovidesonguarantorofguaranteesqualificationandguaranteescapacityforsurveyverified,ledtoguaranteesloanlostauthenticity,andlegitimacy,andeffectivenessof;cycleloanbusinessintheofmortgage.2、橢圓的第二定義:c動(dòng)點(diǎn)M與一個(gè)定點(diǎn)的距離和它到一條定直線的距離的比是常數(shù)時(shí)，eac,,,(0)a這個(gè)點(diǎn)的軌跡是橢圓，定點(diǎn)是橢圓的焦點(diǎn)，定直線叫做橢圓的準(zhǔn)線，常數(shù)e是橢圓的離心率(222xya說明:對于橢圓，相應(yīng)于焦點(diǎn)F(c,0)的準(zhǔn)線方程是(根，,,,1(0)ablx:,22cab2a據(jù)橢圓的對稱性，相應(yīng)于焦點(diǎn)F`(-c,0)的準(zhǔn)線方程是，所以橢圓有兩條準(zhǔn)線(x,,c例題:已知橢圓的焦點(diǎn)是F(-1,0)、F(1,0)，P為橢圓上的一點(diǎn)，且|FF|是|PF|和|PF|的等121212差中項(xiàng).求橢圓的方程;二、雙曲線及其標(biāo)準(zhǔn)方程1、平面內(nèi)與兩個(gè)定點(diǎn)F、F的距離的差的絕對值等于常數(shù)(小于|FF|)的點(diǎn)的軌跡叫做1212雙曲線，這兩個(gè)定點(diǎn)叫做雙曲線的焦點(diǎn)，兩焦點(diǎn)的距離叫做雙曲線的焦距(22xy,,,,1(0,0)ab這個(gè)方程叫做雙曲線的標(biāo)準(zhǔn)方程(它所表示的雙曲線的焦點(diǎn)在x22ab222軸上，焦點(diǎn)是F(-c,0)、F(c,0)，這里c=a+b(同理，焦點(diǎn)在軸上的雙曲線的標(biāo)準(zhǔn)方y(tǒng)1222yx222,,,,1(0,0)ab程是:焦點(diǎn)是F(0，-c)、F(0,c)，這里c=a+b.1222ab橢圓與雙曲線的幾何性質(zhì)橢圓雙曲線方程222cabab,，,,(0,0)a.b.c關(guān)系office,branchoffices(jurisdiction),riskmanagement,marketingmanagementsectorthroughsupervisionandinspectionfoundproblems,shouldbeassignedtheinvestigatorsarecorrectedinatimelymanner.27ththefifthchapterpenaltyunderanyofthefollowingacts,thentherelevantprovisionstopunishtheinvestigators,accordingtotheBank.Toconstituteacrimeshallbeinvestigatedforcriminalresponsibility:(A)onthebusinessthatarenotinvolvedintheinvestigation,issuedasurvey.(B)customercreditinformationarenotthoroughverification.(Iii)toparticipateincreditcustomersurveyisnotinplace,customersanddataisincomplete,untrue;heknowsbearacounterfeitedclientsissuingcredit.(D)doesnotprovideforduediligenceofcreditbusiness,pre-loaninvestigationform,concealingfactsorprovidingfalseinformationorshouldbefoundinanormalinvestigationfailedtodiscovertheriskfactors,leadtothereviewandapprovalpolicyerrors,loanrisk.(Five)onloanguaranteesofsurveynotinplace,notbyprovidesonarrived,andpledgerealforfieldverification,andassessment,andidentificationandregistration,notaccordingtoprovidesonguarantorofguaranteesqualificationandguaranteescapacityforsurveyverified,ledtoguaranteesloanlostauthenticity,andlegitimacy,andeffectivenessof;cycleloanbusinessintheofmortgage趙老師家教圖形x,a,y,bx,a,y,R范圍對稱軸:軸、軸對稱軸:軸、軸對稱性對稱中心:原點(diǎn)對稱中心:原點(diǎn)、、頂點(diǎn)實(shí)軸長、長軸長，短軸長虛軸長離心率，，(0,e,1)有兩條，其方程為漸近線無2、雙曲線的第二定義cee,,(1)當(dāng)點(diǎn)到一個(gè)定點(diǎn)的距離和它到一條定直線的距離的比是常數(shù)時(shí)，這個(gè)a點(diǎn)的軌跡是雙曲線(通常稱為雙曲線的第二定義(定點(diǎn)是雙曲線的焦點(diǎn)，定直線叫做雙曲線的準(zhǔn)線，常數(shù)是雙曲線的離心率(222xyax,,,1對于雙曲線，相應(yīng)于焦點(diǎn)F(c,0)的準(zhǔn)線方程是，根據(jù)雙曲線的對稱22cab2ax,,性，相應(yīng)于焦點(diǎn)F’(-c,0)的準(zhǔn)線方程是，所以雙曲線有兩條準(zhǔn)線(c3office,branchoffices(jurisdiction),riskmanagement,marketingmanagementsectorthroughsupervisionandinspectionfoundproblems,shouldbeassignedtheinvestigatorsarecorrectedinatimelymanner.27ththefifthchapterpenaltyunderanyofthefollowingacts,thentherelevantprovisionstopunishtheinvestigators,accordingtotheBank.Toconstituteacrimeshallbeinvestigatedforcriminalresponsibility:(A)onthebusinessthatarenotinvolvedintheinvestigation,issuedasurvey.(B)customercreditinformationarenotthoroughverification.(Iii)toparticipateincreditcustomersurveyisnotinplace,customersanddataisincomplete,untrue;heknowsbearacounterfeitedclientsissuingcredit.(D)doesnotprovideforduediligenceofcreditbusiness,pre-loaninvestigationform,concealingfactsorprovidingfalseinformationorshouldbefoundinanormalinvestigationfailedtodiscovertheriskfactors,leadtothereviewandapprovalpolicyerrors,loanrisk.(Five)onloanguaranteesofsurveynotinplace,notbyprovidesonarrived,andpledgerealforfieldverification,andassessment,andidentificationandregistration,notaccordingtoprovidesonguarantorofguaranteesqualificationandguaranteescapacityforsurveyverified,ledtoguaranteesloanlostauthenticity,andlegitimacy,andeffectivenessof;cycleloanbusinessintheofmortgage22xy例題:求以橢圓的兩個(gè)頂點(diǎn)為焦點(diǎn)，以橢圓的焦點(diǎn)為頂點(diǎn)的雙曲線方程，并求，,1169此雙曲線的實(shí)軸長、虛軸長、離心率和漸近線方程.三、拋物線及其標(biāo)準(zhǔn)方程1、拋物線的定義:平面內(nèi)與一個(gè)定點(diǎn)F和一條定直線的距離相等的點(diǎn)的軌跡叫做拋物l線(點(diǎn)F叫做拋物線的焦點(diǎn)，定直線叫做拋物線的準(zhǔn)線(l2、一條拋物線，由于它在坐標(biāo)系的位置不同，方程也不同，有四種不同的情況，如下表:圖形標(biāo)準(zhǔn)方程焦點(diǎn)坐標(biāo)準(zhǔn)線方程例1:點(diǎn)M與點(diǎn)到(4，0)的距離比它到直線l:x+5=0的距離小1，求點(diǎn)M的軌跡方程(office,branchoffices(jurisdiction),riskmanagement,marketingmanagementsectorthroughsupervisionandinspectionfoundproblems,shouldbeassignedtheinvestigatorsarecorrectedinatimelymanner.27ththefifthchapterpenaltyunderanyofthefollowingacts,thentherelevantprovisionstopunishtheinvestigators,accordingtotheBank.Toconstituteacrimeshallbeinvestigatedforcriminalresponsibility:(A)onthebusinessthatarenotinvolvedintheinvestigation,issuedasurvey.(B)customercreditinformationarenotthoroughverification.(Iii)toparticipateincreditcustomersurveyisnotinplace,customersanddataisincomplete,untrue;heknowsbearacounterfeitedclientsissuingcredit.(D)doesnotprovideforduediligenceofcreditbusiness,pre-loaninvestigationform,concealingfactsorprovidingfalseinformationorshouldbefoundinanormalinvestigationfailedtodiscovertheriskfactors,leadtothereviewandapprovalpolicyerrors,loanrisk.(Five)onloanguaranteesofsurveynotinplace,notbyprovidesonarrived,andpledgerealforfieldverification,andassessment,andidentificationandregistration,notaccordingtoprovidesonguarantorofguaranteesqualificationandguaranteescapacityforsurveyverified,ledtoguaranteesloanlostauthenticity,andlegitimacy,andeffectivenessof;cycleloanbusinessintheofmortgage趙老師家教2例2:斜率為1的直線經(jīng)過拋物線y=4x的焦點(diǎn)，與拋物線相交于兩點(diǎn)A、B，求線段AB的長.pp2說明:拋物線y=2px(p>0)上一點(diǎn)A(x,y)到焦點(diǎn)的距離這就是拋F(,0)||AFx,，oo022物線的焦半徑公式。焦點(diǎn)弦長|AB|=x+x+p.1223、拋物線y=2px(p>0)的幾何性質(zhì)(1)范圍:因?yàn)閜>0，由方程可知x?0，所以拋物線在y軸的右側(cè)，當(dāng)x的值增大時(shí)，|y|也增大，這說明拋物線向右上方和右下方無限延伸((2)對稱性(3)頂點(diǎn)(4)離心率(5)通徑:過(圓錐曲線)焦點(diǎn)垂直于焦點(diǎn)所在的軸的弦叫做(該圓錐曲線的)通徑.拋物線的通徑長:2p5office,branchoffices(jurisdiction),riskmanagement,marketingmanagementsectorthroughsupervisionandinspectionfoundproblems,shouldbeassignedtheinvestigatorsarecorrectedinatimelymanner.27ththefifthchapterpenaltyunderanyofthefollowingacts,thentherelevantprovisionstopunishtheinvestigators,accordingtotheBank.Toconstituteacrimeshallbeinvestigatedforcriminalresponsibility:(A)onthebusinessthatarenotinvolvedintheinvestigation,issuedasurvey.(B)c