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1、2.2.2橢圓的簡(jiǎn)單幾何性質(zhì)通過(guò)方程,研究平面曲線的性質(zhì)橢圓的定義MF1?MF2?2a(2a?2c?0)ybyMF2Ma圖形F1coF2xoxF1焦點(diǎn)在x軸上焦點(diǎn)在y軸上22xyy2x2標(biāo)準(zhǔn)方程2?2?1?a?b?0?2?2?1?a?b?0?abab焦點(diǎn)坐標(biāo)222a,b,c的關(guān)系a?c?b(a?c?0,a?b?0)焦點(diǎn)位置的判斷橢圓的簡(jiǎn)單幾何性質(zhì)1.范圍2222xyyxx2y2??1??1??022a2b2b2a2x?y?1(a?b?0)a2?b2?1(a?b?0)222xab??12a即得x?a即-a?x?ayb同理,y?b即-
2、b?y?b說(shuō)明橢圓位于直線x=±aox-aa和y=±b所圍成的矩形里-b練習(xí)1.口答下列橢圓的范圍。x2y2??12516?5≤x≤5,?4≤y≤4橢圓的簡(jiǎn)單幾何性質(zhì)2.橢圓的對(duì)稱性22xy??1(a?b?0)22abyox橢圓的簡(jiǎn)單幾何性質(zhì)2.橢圓的對(duì)稱性22xy??1(a?b?0)22abX-X在方程中,把YYx換成---YYx,y方程不變,說(shuō)明:Q(-x,y)P(x,y)橢圓關(guān)于Y軸對(duì)稱;橢圓關(guān)于x軸對(duì)稱;ox橢圓關(guān)于(0,0)點(diǎn)對(duì)稱;N(-x,-y)M(x,-y)坐標(biāo)軸是橢圓的對(duì)稱軸,原點(diǎn)是橢圓的對(duì)稱中心練習(xí)2.下列方程所
3、表示的曲線中,關(guān)于原點(diǎn)對(duì)稱的是(D)22A.x?2yB.y?4x?02222C.x?4y?5xD.9x?y?422xy練習(xí)3:已知點(diǎn)P(3,6)在??1上,則()22abC(A)點(diǎn)(-3,-6)不在橢圓上(B)點(diǎn)(3,-6)不在橢圓上(C)點(diǎn)(-3,6)在橢圓上(D)無(wú)法判斷點(diǎn)(-3,-6),(3,-6),(-3,6)是否在橢圓上橢圓的簡(jiǎn)單幾何性質(zhì)3.頂點(diǎn)與長(zhǎng)短軸22橢圓和它的對(duì)稱軸的xy??1(a?b?0)22四個(gè)交點(diǎn)——橢圓的頂點(diǎn).ab橢圓頂點(diǎn)坐標(biāo)為:A(-a,0)、A(a,0)、y12B2(0,b)B1(0,-b)、B2(0
4、,b)A1(-a,0)A2(a,0)ox回顧:B1(0,-b)焦點(diǎn)坐標(biāo)(±c,0)橢圓的簡(jiǎn)單幾何性質(zhì)長(zhǎng)軸:線段A1A2;長(zhǎng)軸長(zhǎng)
5、A1A2
6、=2a短軸:線段B1B2;短軸長(zhǎng)
7、B1B2
8、=2b焦距
9、F1F2
10、=2c注意yB2(0,b)aA1(-a,0)bA2(a,0)F1aocFx2②a2=b2+c2,
11、B2F2
12、=a;B1(0,-b)練習(xí)4.畫(huà)出下列橢圓的草圖22x2y2xy(1)??1(2)??12516254yyB24433B222A1AA11A212-5-4-3-2--11012345x-5-4-3-2--11012345x-
13、2-2B-3-31-4-4B1小結(jié):由橢圓的范圍、對(duì)稱性和頂點(diǎn),再進(jìn)行描點(diǎn)畫(huà)圖,只須描出較少的點(diǎn),就可以得到較正確的圖形.四、橢圓的離心率c離心率:橢圓的焦距與長(zhǎng)軸長(zhǎng)的比e=,叫做a橢圓的離心率.[1]離心率的取值范圍:因?yàn)閍>c>0,所以014、組中兩個(gè)橢圓的形狀,哪一個(gè)更扁?2222xyxy(1)+=1與??1;9516122222yxy(2)x+=1與??1。2610根據(jù):離心率e越大,橢圓越扁;離心率e越小,橢圓越圓?焦點(diǎn)在y軸上的橢圓的幾何性質(zhì)又如何呢?2222標(biāo)準(zhǔn)方程xyxy??1(a?b?0)??1(a?b?0)2222abba圖象范圍?a?x?a,?b?y?b?b?x?b,?a?y?a對(duì)稱性關(guān)于x軸、y軸成軸對(duì)稱;關(guān)于原點(diǎn)成中心對(duì)稱。頂點(diǎn)坐標(biāo)(?a,0),(0,?b)(?b,0),(0,?a)焦點(diǎn)坐標(biāo)(?c,0)(0,?c)半軸長(zhǎng)長(zhǎng)半軸長(zhǎng)為a,短半軸長(zhǎng)為b.
15、焦距焦距為2c;a,b,c關(guān)系a2=b2+c2離心率ce?(016、注意分清楚焦點(diǎn)的位置,這樣便于直觀地寫(xiě)出a,b的數(shù)值,進(jìn)而求出c,求出橢圓的長(zhǎng)軸和短軸的長(zhǎng)、離心率、焦點(diǎn)和頂點(diǎn)的坐標(biāo)等幾何性質(zhì).變式訓(xùn)練1.求橢圓4x2+9y2=1的長(zhǎng)軸長(zhǎng)和焦距,焦點(diǎn)坐標(biāo),頂點(diǎn)坐標(biāo)和離心率.例2【名師點(diǎn)評(píng)】在求橢圓方程時(shí),要注意根