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1、经典双曲线知识点双曲线:了解双曲线的定义、几何图形和标准方程;了解双曲线的简单几何性质。 重点:双曲线的定义、几何图形和标准方程，以及简单的几何性质. 难点:双曲线的标准方程，双曲线的渐进线. 知识点一:双曲线的定义在平面内，到两个定点、的距离之差的绝对值等于常数(大于0且)的动点的轨迹叫作双曲线.这两个定点、叫双曲线的焦点，两焦点的距离叫作双曲线的焦距. 注意:1. 双曲线的定义中，常数应当满足的约束条件:，这可以借助于三角形中边的相关性质“两边之差小于第三边”来理解; 2. 若去掉定义中的“绝对值”，常数满足约束条件:()，则动点轨迹仅表示双曲线中靠焦点的一支;若()，则动点轨迹仅表示双曲
2、线中靠焦点的一支; 3. 若常数满足约束条件:，则动点轨迹是以F、F为端点的两条射线(包括端点); 124(若常数满足约束条件:，则动点轨迹不存在; 5(若常数，则动点轨迹为线段FF的垂直平分线。 12知识点二:双曲线的标准方程 1(当焦点在轴上时，双曲线的标准方程:，其中; 2(当焦点在轴上时，双曲线的标准方程:，其中. 注意: 1(只有当双曲线的中心为坐标原点，对称轴为坐标轴建立直角坐标系时,才能得到双曲线的标准方程; 2(在双曲线的两种标准方程中，都有; 3(双曲线的焦点总在实轴上，即系数为正的项所对应的坐标轴上.当的系数为正时，焦点在轴上，双曲线的焦点坐标为，;当的系数为正时，焦点在轴
3、上，双曲线的焦点坐标为，. 知识点三:双曲线的简单几何性质 双曲线(a,0，b,0)的简单几何性质 (1)对称性:对于双曲线标准方程(a,0，b,0)，把x换成x，或把y换成y，或把x、y同时换成x、y，方程都不变，所以双曲线(a,0，b,0)是以x轴、y轴为对称轴的轴对称图形，且是以原点为对称中心的中心对称图形，这个对称中心称为双曲线的中心。 (2)范围:双曲线上所有的点都在两条平行直线x=a和x=a的两侧，是无限延伸的。因此双曲线上点的横坐标满足x?-a或x?a。(3)顶点:?双曲线与它的对称轴的交点称为双曲线的顶点。 ?双曲线(a,0，b,0)与坐标轴的两个交点即为双曲线的两个顶点，坐标
4、分别为A(a，0)，A(a，0)，12顶点是双曲线两支上的点中距离最近的点。 ?两个顶点间的线段AA叫作双曲线的实轴;设B(0，b)，B(0，b)为y轴上的两个点，则线段BB叫做双曲线121212的虚轴。实轴和虚轴的长度分别为|AA|=2a，|BB|=2b。a叫做双曲线的实半轴长，b叫做双曲线的虚半轴长。 1212注意:?双曲线只有两个顶点，而椭圆有四个顶点，不能把双曲线的虚轴与椭圆的短轴混淆。 ?双曲线的焦点总在实轴上。?实轴和虚轴等长的双曲线称为等轴双曲线。 appearance of the weld appearance quality technical requirements o
5、f the project must not have a molten metal stream does not melt the base metal to weld, weld seam and heat-affected zone surface must not have cracks, pores, defects such as crater and ash, surface smoothing, weld and base metal should be evenly smooth transition. Width 2-3 mm from the edge of weld
6、Groove. Surface reinforcement should be less than or equal to 1 + 0.2 times the slope edge width, and should not be greater than 4 mm. Depth of undercut should be less than or equal to 0.5 mm, total length of the welds on both sides undercut not exceed 10% of the weld length, and long continuous sho
7、uld not be greater than 100 mm. Wrong side should be less than or at 0.2T, and should not be greater than 2 mm (wall thickness mm t) incomplete or not allow 7.5 7.5.1 installation quality process standards of the electrical enclosure Cabinet surface is clean, neat, no significant phenomenon of conve
8、x, close to nature, close the door. 7.5.2 Cabinet Cabinet face paints no paint, returned to rusted, consistent color. 7.5.3 uniform indirect gap from top to bottom, slot width 1.5mm 7.5.4 adjacent Cabinet surface roughness is 0. 7.5.5 the cabinets firmly fixed, crafts beautiful. 7.5.6 Cabinet surfac
9、e gauge, switch cabinet mark clear, neat, firm paste. 7.5.7 Terminal row of neat, is reliable, the appearance is clean and not damaged. 7.5.8 cables neat and clean, solid binding, binding process in appearance. 7.5.9 the first cable production firm, crafts beautiful, clear signage does not fade. 7.5
10、.10 fireproof plugging tight, no cracks and pores. 7.6 7.6.1 of the standard electrical wiring quality technology cable a, the multi-core wire bunch arrangement should be parallel to each other, horizontal wire harness or wire should be perpendicular to the longitudinal multi-core wire bunch. The di
11、stance between the wire harness and wire harness symmetry, and as close as possible. B-core wiring harness into round, multi-core wire bunch used g wire binding, fastening (4)离心率: ?双曲线的焦距与实轴长的比叫做双曲线的离心率，用e表示，记作。 222 ?因为c,a,0，所以双曲线的离心率。 由c=a+b，可得，所以决定双曲线的开口大小，越大，e也越大，双曲线开口就越开阔。所以离心率可以用来表示双曲线开口的大小程度。?
12、等轴双曲线，所以离心率。 (5)渐近线:经过点A、A作y轴的平行线x=?a，经过点B、B作x轴的平行线y=?b，四条直线围成一个矩形(如图)，2112矩形的两条对角线所在直线的方程是。我们把直线叫做双曲线的渐近线。 注意:双曲线与它的渐近线无限接近，但永不相交。 知识点四:双曲线与的区别和联系 标准方程 图形 焦点 ， ， 焦距 范围 ， ， 对称性 关于x轴、y轴和原点对称 顶点 性质 实轴长=，虚轴长= 轴 离心率 准线方程 渐近线方程 appearance of the weld appearance quality technical requirements of the proje
13、ct must not have a molten metal stream does not melt the base metal to weld, weld seam and heat-affected zone surface must not have cracks, pores, defects such as crater and ash, surface smoothing, weld and base metal should be evenly smooth transition. Width 2-3 mm from the edge of weld Groove. Sur
14、face reinforcement should be less than or equal to 1 + 0.2 times the slope edge width, and should not be greater than 4 mm. Depth of undercut should be less than or equal to 0.5 mm, total length of the welds on both sides undercut not exceed 10% of the weld length, and long continuous should not be
15、greater than 100 mm. Wrong side should be less than or at 0.2T, and should not be greater than 2 mm (wall thickness mm t) incomplete or not allow 7.5 7.5.1 installation quality process standards of the electrical enclosure Cabinet surface is clean, neat, no significant phenomenon of convex, close to
16、 nature, close the door. 7.5.2 Cabinet Cabinet face paints no paint, returned to rusted, consistent color. 7.5.3 uniform indirect gap from top to bottom, slot width 1.5mm 7.5.4 adjacent Cabinet surface roughness is 0. 7.5.5 the cabinets firmly fixed, crafts beautiful. 7.5.6 Cabinet surface gauge, sw
17、itch cabinet mark clear, neat, firm paste. 7.5.7 Terminal row of neat, is reliable, the appearance is clean and not damaged. 7.5.8 cables neat and clean, solid binding, binding process in appearance. 7.5.9 the first cable production firm, crafts beautiful, clear signage does not fade. 7.5.10 firepro
18、of plugging tight, no cracks and pores. 7.6 7.6.1 of the standard electrical wiring quality technology cable a, the multi-core wire bunch arrangement should be parallel to each other, horizontal wire harness or wire should be perpendicular to the longitudinal multi-core wire bunch. The distance betw
19、een the wire harness and wire harness symmetry, and as close as possible. B-core wiring harness into round, multi-core wire bunch used g wire binding, fastening 知识点五:双曲线的渐近线:(1)已知双曲线方程求渐近线方程:若双曲线方程为，则其渐近线方程为注意:(1)已知双曲线方程，将双曲线方程中的“常数”换成“0”，然后因式分解即得渐近线方程。 (2)已知渐近线方程求双曲线方程:若双曲线渐近线方程为，则可设双曲线方程为，根据已知条件，求
20、出即可。(3)与双曲线有公共渐近线的双曲线方程可设为(，焦点在轴上，焦点在y轴上)(4)等轴双曲线的渐近线等轴双曲线的两条渐近线互相垂直，为，因此等轴双曲线可设为. 知识点六:双曲线图像中线段的几何特征: 双曲线，如图: (1)实轴长，虚轴长,焦距， (2)离心率:; (3)顶点到焦点的距离:，; (4)中结合定义与余弦定理，将有关线段、和角结合起来. 1(如何确定双曲线的标准方程,当且仅当双曲线的对称中心在坐标原点，对称轴是坐标轴，双曲线的方程才是标准方程形式。此时，双曲线的焦点在坐标轴上。 2(双曲线标准方程中的三个量a、b、c的几何意义 双曲线标准方程中，a、b、c三个量的大小与坐标系无
21、关，是由双曲线本身所确定的，分别表示双曲线的实半轴长、虚222半轴长和半焦距长，均为正数，且三个量的大小关系为:c,a，c,b，且c=b+a。 3(如何由双曲线标准方程判断焦点位置 222 双曲线的焦点总在实轴上，因此已知标准方程，判断焦点位置的方法是:看x、y的系数，如果x项的系数是正的，那2么焦点在x轴上;如果y项的系数是正的，那么焦点在y轴上。注意:对于双曲线，a不一定大于b，因此不能像椭圆那样通过比较分母的大小来判定焦点在哪一条坐标轴上。 224(方程Ax+By=C(A、B、C均不为零)表示双曲线的条件 appearance of the weld appearance quality
22、 technical requirements of the project must not have a molten metal stream does not me 3 / 8 lt the base metal to weld, weld seam and heat-affected zone surface must not have cracks, pores, defects such as crater and ash, surface smoothing, weld and base metal should be evenly smooth transition. Wid
23、th 2-3 mm from the edge of weld Groove. Surface reinforcement should be less than or equal to 1 + 0.2 times the slope edge width, and should not be greater than 4 mm. Depth of undercut should be less than or equal to 0.5 mm, total length of the welds on both sides undercut not exceed 10% of the weld
24、 length, and long continuous should not be greater than 100 mm. Wrong side should be less than or at 0.2T, and should not be greater than 2 mm (wall thickness mm t) incomplete or not allow 7.5 7.5.1 installation quality process standards of the electrical enclosure Cabinet surface is clean, neat, no
25、 significant phenomenon of convex, close to nature, close the door. 7.5.2 Cabinet Cabinet face paints no paint, returned to rusted, consistent color. 7.5.3 uniform indirect gap from top to bottom, slot width 1.5mm 7.5.4 adjacent Cabinet surface roughness is 0. 7.5.5 the cabinets firmly fixed, crafts
26、 beautiful. 7.5.6 Cabinet surface gauge, switch cabinet mark clear, neat, firm paste. 7.5.7 Terminal row of neat, is reliable, the appearance is clean and not damaged. 7.5.8 cables neat and clean, solid binding, binding process in appearance. 7.5.9 the first cable production firm, crafts beautiful,
27、clear signage does not fade. 7.5.10 fireproof plugging tight, no cracks and pores. 7.6 7.6.1 of the standard electrical wiring quality technology cable a, the multi-core wire bunch arrangement should be parallel to each other, horizontal wire harness or wire should be perpendicular to the longitudin
28、al multi-core wire bunch. The distance between the wire harness and wire harness symmetry, and as close as possible. B-core wiring harness into round, multi-core wire bunch used g wire binding, fastening 22 方程Ax+By=C可化为，即，所以只有A、B异号，方程表示双曲线。当时，双曲线的焦点在x轴上;当时，双曲线的焦点在y轴上。 5(求双曲线标准方程的常用方法: ?待定系数法:由题目条件确定
29、焦点的位置，从而确定方程的类型，设出标准方程，再由条件确定方程中的参数、的值。其主要步骤是“先定型，再定量”; ?定义法:由题目条件判断出动点的轨迹是什么图形，然后再根据定义确定方程。 注意:若定义中“差的绝对值”中的绝对值去掉，点的集合成为双曲线的一支，先确定方程类型，再确定参数a、b，即先定型，再定量。若两种类型都有可能，则需分类讨论。 6(如何解决与焦点三角形?PFF(P为双曲线上的点)有关的计算问题, 12与焦点三角形有关的计算问题时，常考虑到用双曲线的定义及余弦定理(或勾股定理)、三角形面积公式相结合的方法进行计算与解题，将有关线段、，有关角结合起来，建立、之间的关系. 222(如何
30、确定离心率e的取值情况与双曲线形状的关系, :离心率，因为c=a+b，用a、b表示为，7当e越大时，越大，即渐近线夹角(含x轴)越大，故开口越大;反之，e越小，开口越小。离心率反映了双曲线开口的大小，且e,1。 (椭圆、双曲线的区别和联系: 8椭圆 双曲线 根据|MF|+|MF|=2a 根据|MF|,|MF|=?2a 1212a,c,0， 0,a,c， 222222a,c=b(b,0) c,a=b(b,0) ， ， (a,b,0) (a,0，b,0，a不一定大于b) 标准方程统一为: 2222类型一:双曲线的定义1(已知?O:(x+5)+y=4，?O:(x,5)+y=9 12(1)若动圆P与?
31、，?均内切，求动圆圆心P点的轨迹;(2)若动圆Q与?，?均外切，求动圆圆心Q点的轨迹。 1212解析:(1)设?P半径为R， ?O与?O相离， ?|PO|=R,2，|PO|=R,3 ?|PO|,|PO|=1，又|OO|=10 12121212?由双曲线的定义，P点的轨迹是以O，O为焦点，2a=1，2c,10的双曲线的右支。 12appearance of the weld appearance quality technical requirements of the project must not have a molten metal stream does not melt the b
32、ase metal to weld, weld seam and heat-affected zone surface must not have cracks, pores, defects such as crater and ash, surface smoothing, weld and base metal should be evenly smooth transition. Width 2-3 mm from the edge of weld Groove. Surface reinforcement should be less than or equal to 1 + 0.2
33、 times the slope edge width, and should not be greater than 4 mm. Depth of undercut should be less than or equal to 0.5 mm, total length of the welds on both sides undercut not exceed 10% of the weld length, and long continuous should not be greater than 100 mm. Wrong side should be less than or at
34、0.2T, and should not be greater than 2 mm (wall thickness mm t) incomplete or not allow 7.5 7.5.1 installation quality process standards of the electrical enclosure Cabinet surface is clean, neat, no significant phenomenon of convex, close to nature, close the door. 7.5.2 Cabinet Cabinet face paints
35、 no paint, returned to rusted, consistent color. 7.5.3 uniform indirect gap from top to bottom, slot width 1.5mm 7.5.4 adjacent Cabinet surface roughness is 0. 7.5.5 the cabinets firmly fixed, crafts beautiful. 7.5.6 Cabinet surface gauge, switch cabinet mark clear, neat, firm paste. 7.5.7 Terminal
36、row of neat, is reliable, the appearance is clean and not damaged. 7.5.8 cables neat and clean, solid binding, binding process in appearance. 7.5.9 the first cable production firm, crafts beautiful, clear signage does not fade. 7.5.10 fireproof plugging tight, no cracks and pores. 7.6 7.6.1 of the s
37、tandard electrical wiring quality technology cable a, the multi-core wire bunch arrangement should be parallel to each other, horizontal wire harness or wire should be perpendicular to the longitudinal multi-core wire bunch. The distance between the wire harness and wire harness symmetry, and as clo
38、se as possible. B-core wiring harness into round, multi-core wire bunch used g wire binding, fastening (2)设?Q半径为r，则|QO|=r+2，|QO|=r+3 ?|QO|,|QO|=1，又|OO|=10 122112?由双曲线的定义，Q点的轨迹是以O，O为焦点，2a=1，2c,10的双曲线的左支。 12举一反三:【变式1】已知定点F(,2,0)、F(2,0)，平面内满足下列条件的动点P的轨迹为双曲线的是( )A(|PF|,|PF|=121222?3B(|PF|,|PF|=?4C(|PF|,
39、|PF|=?5 D(|PF|,|PF|=?4 【答案】A 121212【变式2】已知点F(0,13)、F(0,13)，动点P到F与F的距离之差的绝对值为26，则动点P的轨迹方程为( ) 1212A(y=0 B(y=0(x?,13或x?13)C(x=0(|y|?13)D(以上都不对【答案】C 【变式3】已知点P(x,y)的坐标满足，则动点P的轨迹是( ) A(椭圆 B(双曲线中的一支 C(两条射线 D(以上都不对 答案:B 类型二:双曲线的标准方程: 2(求与双曲线有公共焦点，且过点的双曲线的标准方程。 解法一: 依题意设双曲线方程为,=1由已知得，又双曲线过点，? :故所求双曲线的方程为.解法
40、二:依题意设双曲线方程为，将点代入，解得，所以双曲线方程为.【变式1】求与椭圆有共同的焦点，且过点的双曲线的标准方程。【答案】依题意设双曲线方程为 由已知得，又，? ? 故所求双曲线的方程为. 双曲线过点【变式2】求中心在原点，对称轴为坐标轴，且顶点在轴，焦距为10，的双曲线的标准方程.【答案】 3(已知双曲线的两个焦点F、F之间的距离为26，双曲线上一点到两焦点的距离之差的绝对值为24，求双曲线的标准方程。 12222解析:由题意得2a=24，2c=26。?a=12，c=13，b=13,12=25。 当双曲线的焦点在x轴上时，双曲线的方程为; 当双曲线的焦点在y轴上时，双曲线的方程为。 22
41、 总结升华:求双曲线的标准方程就是求a、b的值，同时还要确定焦点所在的坐标轴。双曲线所在的坐标轴，不像椭圆2222那样看x、y的分母的大小，而是看x、y的系数的正负。 【变式】求中心在原点，对称轴为坐标轴，且虚轴长与实轴长的比为，焦距为10的双曲线的标准方程.【答案】由已知设, ，则() 依题意，解得. ?当双曲线的焦点在x轴上时，双曲线的方程为 当双曲线的焦点在y轴上时，双曲线的方程为. 类型三:双曲线的几何性质 4(方程表示双曲线，求实数m的取值范围。 appearance of the weld appearance quality technical requirements of t
42、he project must not have a molten metal stream does not me 5 / 8 lt the base metal to weld, weld seam and heat-affected zone surface must not have cracks, pores, defects such as crater and ash, surface smoothing, weld and base metal should be evenly smooth transition. Width 2-3 mm from the edge of w
43、eld Groove. Surface reinforcement should be less than or equal to 1 + 0.2 times the slope edge width, and should not be greater than 4 mm. Depth of undercut should be less than or equal to 0.5 mm, total length of the welds on both sides undercut not exceed 10% of the weld length, and long continuous
44、 should not be greater than 100 mm. Wrong side should be less than or at 0.2T, and should not be greater than 2 mm (wall thickness mm t) incomplete or not allow 7.5 7.5.1 installation quality process standards of the electrical enclosure Cabinet surface is clean, neat, no significant phenomenon of c
45、onvex, close to nature, close the door. 7.5.2 Cabinet Cabinet face paints no paint, returned to rusted, consistent color. 7.5.3 uniform indirect gap from top to bottom, slot width 1.5mm 7.5.4 adjacent Cabinet surface roughness is 0. 7.5.5 the cabinets firmly fixed, crafts beautiful. 7.5.6 Cabinet su
46、rface gauge, switch cabinet mark clear, neat, firm paste. 7.5.7 Terminal row of neat, is reliable, the appearance is clean and not damaged. 7.5.8 cables neat and clean, solid binding, binding process in appearance. 7.5.9 the first cable production firm, crafts beautiful, clear signage does not fade.
47、 7.5.10 fireproof plugging tight, no cracks and pores. 7.6 7.6.1 of the standard electrical wiring quality technology cable a, the multi-core wire bunch arrangement should be parallel to each other, horizontal wire harness or wire should be perpendicular to the longitudinal multi-core wire bunch. Th
48、e distance between the wire harness and wire harness symmetry, and as close as possible. B-core wiring harness into round, multi-core wire bunch used g wire binding, fastening 解析:由题意得或或 。?实数m的取值范围为。 22总结升华:方程Ax+By=1表示双曲线时，A、B异号。 【变式1】k,9是方程表示双曲线的( ) A(充分必要条件 B(充分不必要条件 C(必要不充分条件 D(既不充分又不必要条件 【答案】B 【变式2】求双曲线的焦距。 【答案】8 22 【变式3】已知双曲线8kx,ky=2的一个焦点为，则k的值等于( ) A(,2 B(1 C(,1 D(【答案】C 【变式4】(2011 湖南)设双曲线的渐近线方程为，则的值为 A(4 B(3 C(2 D(1【答案】C 5(已知双曲线方程，求渐近线方程。 (1);(2);(3);(4) 解析:(1)双曲线的渐近线方程为:即(2)双曲线的渐近线方程为:即(3)双曲线的渐近线方程为: 即 (4)双曲线的渐近线方程为:即 总结升华:双曲线的渐近线方程为，双
