高中数学圆锥曲线基础知识点

高中数学圆锥曲线基础知识点一、椭圆及其标准方程1、椭圆的定义:平面内到两定点F、F的距离之和等于常数(大于|FF|)的点的轨迹叫做椭1212圆(这两个定点叫做椭圆的焦点，两焦点的距离叫做焦距(两种标准方程的比较22xy(1)表示焦点在x轴上的椭圆，焦点是F(-c,0),F(c,0);，,,,1(0)ab1222ab22yx(2)表示焦点在y轴上的椭圆，焦点是F(0,-c),F(0,c);，,,,1(0)ab1222ab在两种标准方程中2221a,b,c的关系c=a-b不变，只须将(1)方程的x、y互换即可得到(2);?222?a,b，?可以根据分母的大小来判定焦点在哪一个坐标轴上(?ce,离心率定义:椭圆的焦距与长轴长的比，叫做椭圆的离心率(离心率e的取值范围:a0<e<1(方程图形范围，，对称性关于轴、轴、坐标原点对称关于轴、轴、坐标原点对称A(-a,0)，A(a,0)(0,-a)，A(0,a)A1212顶点B(0,-b)，B(0,b)B(-b,0)，B(b,0)1212长轴长=2a,短轴长=2b离心率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、椭圆的第二定义:c动点M与一个定点的距离和它到一条定直线的距离的比是常数时，eac,,,(0)a这个点的轨迹是椭圆，定点是椭圆的焦点，定直线叫做椭圆的准线，常数e是椭圆的离心率(222xya说明:对于椭圆，相应于焦点F(c,0)的准线方程是(根，,,,1(0)ablx:,22cab2a据椭圆的对称性，相应于焦点F`(-c,0)的准线方程是，所以椭圆有两条准线(x,,c例题:已知椭圆的焦点是F(-1,0)、F(1,0)，P为椭圆上的一点，且|FF|是|PF|和|PF|的等121212差中项.求椭圆的方程;二、双曲线及其标准方程1、平面内与两个定点F、F的距离的差的绝对值等于常数(小于|FF|)的点的轨迹叫做1212双曲线，这两个定点叫做双曲线的焦点，两焦点的距离叫做双曲线的焦距(22xy,,,,1(0,0)ab这个方程叫做双曲线的标准方程(它所表示的双曲线的焦点在x22ab222轴上，焦点是F(-c,0)、F(c,0)，这里c=a+b(同理，焦点在轴上的双曲线的标准方y1222yx222,,,,1(0,0)ab程是:焦点是F(0，-c)、F(0,c)，这里c=a+b.1222ab椭圆与双曲线的几何性质椭圆双曲线方程222cabab,，,,(0,0)a.b.c关系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赵老师家教图形x,a,y,bx,a,y,R范围对称轴:轴、轴对称轴:轴、轴对称性对称中心:原点对称中心:原点、、顶点实轴长、长轴长，短轴长虚轴长离心率，，(0,e,1)有两条，其方程为渐近线无2、双曲线的第二定义cee,,(1)当点到一个定点的距离和它到一条定直线的距离的比是常数时，这个a点的轨迹是双曲线(通常称为双曲线的第二定义(定点是双曲线的焦点，定直线叫做双曲线的准线，常数是双曲线的离心率(222xyax,,,1对于双曲线，相应于焦点F(c,0)的准线方程是，根据双曲线的对称22cab2ax,,性，相应于焦点F’(-c,0)的准线方程是，所以双曲线有两条准线(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例题:求以椭圆的两个顶点为焦点，以椭圆的焦点为顶点的双曲线方程，并求，,1169此双曲线的实轴长、虚轴长、离心率和渐近线方程.三、抛物线及其标准方程1、抛物线的定义:平面内与一个定点F和一条定直线的距离相等的点的轨迹叫做抛物l线(点F叫做抛物线的焦点，定直线叫做抛物线的准线(l2、一条抛物线，由于它在坐标系的位置不同，方程也不同，有四种不同的情况，如下表:图形标准方程焦点坐标准线方程例1:点M与点到(4，0)的距离比它到直线l:x+5=0的距离小1，求点M的轨迹方程(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赵老师家教2例2:斜率为1的直线经过抛物线y=4x的焦点，与抛物线相交于两点A、B，求线段AB的长.pp2说明:抛物线y=2px(p>0)上一点A(x,y)到焦点的距离这就是抛F(,0)||AFx,，oo022物线的焦半径公式。焦点弦长|AB|=x+x+p.1223、抛物线y=2px(p>0)的几何性质(1)范围:因为p>0，由方程可知x?0，所以抛物线在y轴的右侧，当x的值增大时，|y|也增大，这说明抛物线向右上方和右下方无限延伸((2)对称性(3)顶点(4)离心率(5)通径:过(圆锥曲线)焦点垂直于焦点所在的轴的弦叫做(该圆锥曲线的)通径.抛物线的通径长:2p5office,branchoffices(jurisdiction),riskmanagement,marketingmanagementsectorthroughsupervisionandinspectionfoundproblems,shouldbeassignedtheinvestigatorsarecorrectedinatimelymanner.27ththefifthchapterpenaltyunderanyofthefollowingacts,thentherelevantprovisionstopunishtheinvestigators,accordingtotheBank.Toconstituteacrimeshallbeinvestigatedforcriminalresponsibility:(A)onthebusinessthatarenotinvolvedintheinvestigation,issuedasurvey.(B)c