2.3.2双曲线的简单几何性质(一)方程焦点a.b.c的关系图象定义||MF1|-|MF2||=2a（０<2a<|F1F2|）F(±c,0)F(0,±c)渐近线离心率顶点对称性范围|x|?a,|y|≤b|x|≥a，y?R对称轴：x轴，y轴对称中心：原点对称轴：x轴，y轴对称中心：原点（-a,0)(a,0)(0,b)(0,-b)长轴：2a短轴：2b(-a,0)(a,0)实轴：2a虚轴：2be=ac(0＜e＜1)ace=(e?1)无y=abx±yXF10F2M图象e越接近1越扁，e越接近0越圆e越大，双曲线张口越大2、对称性一、研究双曲线的简单几何性质1、范围关于x轴、y轴和原点都是对称。x轴、y轴是双曲线的对称轴，原点是对称中心，又叫做双曲线的中心。xyo-aa(-x,-y)(-x,y)(x,y)(x,-y)课堂新授3、顶点（1）双曲线与对称轴的交点，叫做双曲线的顶点xyo-bb-aa如图，线段叫做双曲线的实轴，它的长为2a,a叫做实半轴长；线段叫做双曲线的虚轴，它的长为2b,b叫做双曲线的虚半轴长（2）实轴与虚轴等长的双曲线叫等轴双曲线（3）xyoa4、渐近线MNP(2)等轴双曲线的渐近线？（3）利用渐近线可以较准确的画出双曲线的草图不相交渐近线5、离心率e反映了双曲线开口大小：e越大双曲线开口越大e越小双曲线开口越小xyo（2）离心率的几何意义：ab（4）等轴双曲线的离心率e=?（3）离心率范围：e>1XYF1F2OB1B2A2A1焦点在y轴上的双曲线图像xyo-aab-b（1）范围:（2）对称性:关于x轴、y轴、原点都对称（3）顶点:(0,-a)、(0,a)（4）渐近线:（5）离心率:记忆或或关于坐标轴和原点都对称性质双曲线范围对称性顶点渐近线离心率图象例1:求双曲线的实半轴长,虚半轴长,焦点坐标,离心率.渐近线方程。解：把方程化为标准方程可得:实半轴长a=4虚半轴长b=3半焦距c=焦点坐标是(0,-5),(0,5)离心率:渐近线方程:14416922=-xy1342222=-xy53422=+45==ace例题讲解例2:求双曲线的实半轴长、虚轴长、焦点坐标、顶点坐标、离心率、渐近线方程.解:由题意可得实半轴长:虚轴长:焦点坐标:离心率:渐近线方程:练习：a=2顶点坐标:(-2,0)，(2,0)请你写出一个以为渐近线的双曲线方程.你能写出所有以为渐近线的双曲线方程吗?练习:求下列双曲线的标准方程：法二：巧设方程,运用待定系数法.⑴设双曲线方程为,法二：设双曲线方程为∴双曲线方程为∴,解之得k=4,关于x轴、y轴、原点对称图形方程范围对称性顶点离心率A1（－a，0），A2（a，0）A1（0，－a），A2（0，a）关于x轴、y轴、原点对称渐近线..yB2A1A2B1xOF2F1xB1yO.F2F1B2A1A2.F1(-c,0)F2(c,0)F2(0,c)F1(0,-c)ThankYou!能不能直接由双曲线方程推出渐近线方程？双曲线方程中，把1改为0，得
⑴与双曲线有共同渐近线，且过点；
⑵与双曲线有公共焦点，且过点
⑴法一:直接设标准方程,运用待定系数法考虑.(一般要分类讨论)
解:双曲线的渐近线为,令x=-3,y=±4,因,
故点在射线（x≤0）及x轴负半轴之间,
∴双曲线焦点在x轴上,∴设双曲线方程为(a>0,b>0),
∴解之得,∴双曲线方程为
⑴与双曲线有共同渐近线，且过点；
法一:直接设标准方程,运用待定系数法
⑵解:设双曲线方程为(a>0,b>0)
则解之得
∴双曲线方程为
根据下列条件，求双曲线方程:
⑵与双曲线有公共焦点，且过点.
Sheet3
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x=
y=4(1+(1/9)x2)1/2
y=-4(1+(1/9)x2)1/2
y=4(-1+(1/9)x2)1/2
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