圆锥曲线的所有定义,性质! 圆锥曲线的统一定义: 其他性质:(焦点△,焦半径,S焦点,通...
\u6c42\u5706\u9525\u66f2\u7ebf\u6240\u6709\u7684\u6027\u8d28\u5706\u9525\u66f2\u7ebf \u5706\u9525\u66f2\u7ebf\u5305\u62ec\u692d\u5706\uff0c\u53cc\u66f2\u7ebf\uff0c\u629b\u7269\u7ebf
1. \u692d\u5706\uff1a\u5230\u4e24\u4e2a\u5b9a\u70b9\u7684\u8ddd\u79bb\u4e4b\u548c\u7b49\u4e8e\u5b9a\u957f\uff08\u5b9a\u957f\u5927\u4e8e\u4e24\u4e2a\u5b9a\u70b9\u95f4\u7684\u8ddd\u79bb\uff09\u7684\u52a8\u70b9\u7684\u8f68\u8ff9\u53eb\u505a\u692d\u5706\u3002\u5373\uff1a{P| |PF1|+|PF2|=2a, (2a>|F1F2|)}\u3002
2. \u53cc\u66f2\u7ebf\uff1a\u5230\u4e24\u4e2a\u5b9a\u70b9\u7684\u8ddd\u79bb\u7684\u5dee\u7684\u7edd\u5bf9\u503c\u4e3a\u5b9a\u503c\uff08\u5b9a\u503c\u5c0f\u4e8e\u4e24\u4e2a\u5b9a\u70b9\u7684\u8ddd\u79bb\uff09\u7684\u52a8\u70b9\u8f68\u8ff9\u53eb\u505a\u53cc\u66f2\u7ebf\u3002\u5373{P|||PF1|-|PF2||=2a, (2a<|F1F2|)}\u3002
3. \u629b\u7269\u7ebf\uff1a\u5230\u4e00\u4e2a\u5b9a\u70b9\u548c\u4e00\u6761\u5b9a\u76f4\u7ebf\u7684\u8ddd\u79bb\u76f8\u7b49\u7684\u52a8\u70b9\u8f68\u8ff9\u53eb\u505a\u629b\u7269\u7ebf\u3002
4. \u5706\u9525\u66f2\u7ebf\u7684\u7edf\u4e00\u5b9a\u4e49\uff1a\u5230\u5b9a\u70b9\u7684\u8ddd\u79bb\u4e0e\u5230\u5b9a\u76f4\u7ebf\u7684\u8ddd\u79bb\u7684\u6bd4e\u662f\u5e38\u6570\u7684\u70b9\u7684\u8f68\u8ff9\u53eb\u505a\u5706\u9525\u66f2\u7ebf\u3002\u5f5301\u65f6\u4e3a\u53cc\u66f2\u7ebf\u3002
\u00b7\u5706\u9525\u66f2\u7ebf\u7531\u6765\uff1a\u5706\uff0c\u692d\u5706\uff0c\u53cc\u66f2\u7ebf\uff0c\u629b\u7269\u7ebf\u540c\u5c5e\u4e8e\u5706\u9525\u66f2\u7ebf\u3002\u65e9\u5728\u4e24\u5343\u591a\u5e74\u524d\uff0c\u53e4\u5e0c\u814a\u6570\u5b66\u5bb6\u5bf9\u5b83\u4eec\u5df2\u7ecf\u5f88\u719f\u6089\u4e86\u3002\u53e4\u5e0c\u814a\u6570\u5b66\u5bb6\u963f\u6ce2\u7f57\u5c3c\u91c7\u7528\u5e73\u9762\u5207\u5272\u5706\u9525\u7684\u65b9\u6cd5\u6765\u7814\u7a76\u8fd9\u51e0\u79cd\u66f2\u7ebf\u3002\u7528\u5782\u76f4\u4e0e\u9525\u8f74\u7684\u5e73\u9762\u53bb\u622a\u5706\u9525\uff0c\u5f97\u5230\u7684\u662f\u5706\uff1b\u628a\u5e73\u9762\u6e10\u6e10\u503e\u659c\uff0c\u5f97\u5230\u692d\u5706\uff1b\u5f53\u5e73\u9762\u548c\u5706\u9525\u7684\u4e00\u6761\u6bcd\u7ebf\u5e73\u884c\u65f6\uff0c\u5f97\u5230\u629b\u7269\u7ebf\uff1b\u5f53\u5e73\u9762\u518d\u503e\u659c\u4e00\u4e9b\u5c31\u53ef\u4ee5\u5f97\u5230\u53cc\u66f2\u7ebf\u3002\u963f\u6ce2\u7f57\u5c3c\u66fe\u628a\u692d\u5706\u53eb\u201c\u4e8f\u66f2\u7ebf\u201d\uff0c\u628a\u53cc\u66f2\u7ebf\u53eb\u505a\u201c\u8d85\u66f2\u7ebf\u201d\uff0c\u628a\u629b\u7269\u7ebf\u53eb\u505a\u201c\u9f50\u66f2\u7ebf\u201d\u3002
\u00b7\u5706\u9525\u66f2\u7ebf\u7684\u53c2\u6570\u65b9\u7a0b\u548c\u76f4\u89d2\u5750\u6807\u65b9\u7a0b\uff1a
1\uff09\u692d\u5706
\u53c2\u6570\u65b9\u7a0b\uff1ax=X+acos\u03b8 y=Y+bsin\u03b8 (\u03b8\u4e3a\u53c2\u6570 )
\u76f4\u89d2\u5750\u6807\uff08\u4e2d\u5fc3\u4e3a\u539f\u70b9\uff09\uff1ax^2/a^2 + y^2/b^2 = 1
2\uff09\u53cc\u66f2\u7ebf
\u53c2\u6570\u65b9\u7a0b\uff1ax=X+asec\u03b8 y=Y+btan\u03b8 (\u03b8\u4e3a\u53c2\u6570 )
\u76f4\u89d2\u5750\u6807\uff08\u4e2d\u5fc3\u4e3a\u539f\u70b9\uff09\uff1ax^2/a^2 - y^2/b^2 = 1 (\u5f00\u53e3\u65b9\u5411\u4e3ax\u8f74\uff09 y^2/a^2 - x^2/b^2 = 1 (\u5f00\u53e3\u65b9\u5411\u4e3ay\u8f74\uff09
3\uff09\u629b\u7269\u7ebf
\u53c2\u6570\u65b9\u7a0b\uff1ax=2pt^2 y=2pt (t\u4e3a\u53c2\u6570)
\u76f4\u89d2\u5750\u6807\uff1ay=ax^2+bx+c (\u5f00\u53e3\u65b9\u5411\u4e3ay\u8f74, a0 \uff09 x=ay^2+by+c \uff08\u5f00\u53e3\u65b9\u5411\u4e3ax\u8f74, a0 )
\u5706\u9525\u66f2\u7ebf\uff08\u4e8c\u6b21\u975e\u5706\u66f2\u7ebf\uff09\u7684\u7edf\u4e00\u6781\u5750\u6807\u65b9\u7a0b\u4e3a
\u03c1=ep/(1-e\u00d7cos\u03b8)
\u5176\u4e2de\u8868\u793a\u79bb\u5fc3\u7387\uff0cp\u4e3a\u7126\u70b9\u5230\u51c6\u7ebf\u7684\u8ddd\u79bb\u3002
\u7126\u70b9\u5230\u6700\u8fd1\u7684\u51c6\u7ebf\u7684\u8ddd\u79bb\u7b49\u4e8eex\u00b1a
\u3002\u5706\u9525\u66f2\u7ebf\u7684\u7126\u534a\u5f84\uff08\u7126\u70b9\u5728x\u8f74\u4e0a\uff0cF1 F2\u4e3a\u5de6\u53f3\u7126\u70b9\uff0cP\uff08x\uff0cy\uff09\uff0c\u957f\u534a\u8f74\u957f\u4e3aa\uff09
\u692d\u5706\uff1a\u692d\u5706\u4e0a\u4efb\u4e00\u70b9\u548c\u7126\u70b9\u7684\u8fde\u7ebf\u6bb5\u7684\u957f\u79f0\u4e3a\u7126\u534a\u5f84\u3002
|PF1|=a+ex |PF2|=a-ex
\u53cc\u66f2\u7ebf\uff1a
P\u5728\u5de6\u652f\uff0c|PF1|=\uff0da-ex |PF2|=a-ex
P\u5728\u53f3\u652f\uff0c|PF1|=a+ex |PF2|=\uff0da+ex
P\u5728\u4e0b\u652f\uff0c|PF1|= \uff0da-ey |PF2|=a-ey
P\u5728\u4e0a\u652f\uff0c|PF1|= a+ey |PF2|=\uff0da+ey
\u5706\u9525\u66f2\u7ebf\u7684\u5149\u5b66\u6027\u8d28\uff1a
1\uff09\u692d\u5706\uff1a\u70b9\u5149\u6e90\u5728\u4e00\u4e2a\u7126\u70b9\u4e0a\uff0c\u5149\u7ebf\u901a\u8fc7\u53e6\u4e00\u4e2a\u7126\u70b9\u3002
2\uff09\u53cc\u66f2\u7ebf\uff1a\u70b9\u5149\u6e90\u5728\u4e00\u4e2a\u7126\u70b9\u4e0a\uff0c\u53cd\u5c04\u5149\u7ebf\u4e0e\u53e6\u4e00\u7126\u70b9\u5230\u53cd\u5c04\u70b9\u7684\u8fde\u7ebf\u5728\u540c\u4e00\u6761\u76f4\u7ebf\u4e0a\u3002
3\uff09\u629b\u7269\u7ebf\uff1a\u70b9\u5149\u6e90\u5728\u7126\u70b9\u4e0a\uff0c\u53cd\u5c04\u5149\u7ebf\u76f8\u4e92\u5e73\u884c\u4e14\u5782\u76f4\u4e8e\u51c6\u7ebf\u3002\u5177\u4f53\u5e94\u7528\uff1a\u63a2\u7167\u706f\u3002
[\u7f16\u8f91\u672c\u6bb5]\u5706\u9525\u66f2\u7ebf\u4e2d\u6c42\u70b9\u7684\u8f68\u8ff9\u65b9\u7a0b
\u5728\u6c42\u66f2\u7ebf\u7684\u8f68\u8ff9\u65b9\u7a0b\u65f6\uff0c\u5982\u679c\u80fd\u591f\u5c06\u9898\u8bbe\u6761\u4ef6\u8f6c\u5316\u4e3a\u5177\u6709\u67d0\u79cd\u52a8\u611f\u7684\u76f4\u89c2\u56fe\u5f62\uff0c\u901a\u8fc7\u89c2\u5bdf\u56fe\u5f62\u7684\u53d8\u5316\u8fc7\u7a0b\uff0c\u53d1\u73b0\u5176\u5185\u5728\u8054\u7cfb\uff0c\u627e\u51fa\u54ea\u4e9b\u662f\u53d8\u5316\u7684\u91cf\uff08\u6216\u5173\u7cfb\uff09\u3001\u54ea\u4e9b\u662f\u59cb\u7ec8\u4fdd\u6301\u4e0d\u53d8\u7684\u91cf\uff08\u6216\u5173\u7cfb\uff09\uff0c\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u4ece\u627e\u51fa\u7684\u4e0d\u53d8\u91cf\uff08\u6216\u5173\u7cfb\uff09\u51fa\u53d1\uff0c\u6253\u5f00\u89e3\u9898\u601d\u8def\uff0c\u786e\u5b9a\u89e3\u9898\u65b9\u6cd5
\u51e0\u4f55\u89c2\u70b9
\u7528\u4e00\u4e2a\u5e73\u9762\u53bb\u622a\u4e00\u4e2a\u5706\u9525\u9762\uff0c\u5f97\u5230\u7684\u4ea4\u7ebf\u5c31\u79f0\u4e3a\u5706\u9525\u66f2\u7ebf\uff08conic sections)\u3002
\u901a\u5e38\u63d0\u5230\u7684\u5706\u9525\u66f2\u7ebf\u5305\u62ec\u692d\u5706\uff0c\u53cc\u66f2\u7ebf\u548c\u629b\u7269\u7ebf\uff0c\u4f46\u4e25\u683c\u6765\u8bb2\uff0c\u5b83\u8fd8\u5305\u62ec\u4e00\u4e9b\u9000\u5316\u60c5\u5f62\u3002\u5177\u4f53\u800c\u8a00\uff1a
1) \u5f53\u5e73\u9762\u4e0e\u5706\u9525\u9762\u7684\u6bcd\u7ebf\u5e73\u884c\uff0c\u4e14\u4e0d\u8fc7\u5706\u9525\u9876\u70b9\uff0c\u7ed3\u679c\u4e3a\u629b\u7269\u7ebf\u3002
2) \u5f53\u5e73\u9762\u4e0e\u5706\u9525\u9762\u7684\u6bcd\u7ebf\u5e73\u884c\uff0c\u4e14\u8fc7\u5706\u9525\u9876\u70b9\uff0c\u7ed3\u679c\u9000\u5316\u4e3a\u4e00\u6761\u76f4\u7ebf\u3002
3) \u5f53\u5e73\u9762\u53ea\u4e0e\u5706\u9525\u9762\u4e00\u4fa7\u76f8\u4ea4\uff0c\u4e14\u4e0d\u8fc7\u5706\u9525\u9876\u70b9\uff0c\u7ed3\u679c\u4e3a\u692d\u5706\u3002
4) \u5f53\u5e73\u9762\u53ea\u4e0e\u5706\u9525\u9762\u4e00\u4fa7\u76f8\u4ea4\uff0c\u4e14\u4e0d\u8fc7\u5706\u9525\u9876\u70b9\uff0c\u5e76\u4e0e\u5706\u9525\u9762\u7684\u5bf9\u79f0\u8f74\u5782\u76f4\uff0c\u7ed3\u679c\u4e3a\u5706\u3002
5) \u5f53\u5e73\u9762\u53ea\u4e0e\u5706\u9525\u9762\u4e00\u4fa7\u76f8\u4ea4\uff0c\u4e14\u8fc7\u5706\u9525\u9876\u70b9\uff0c\u7ed3\u679c\u9000\u5316\u4e3a\u4e00\u4e2a\u70b9\u3002
6) \u5f53\u5e73\u9762\u4e0e\u5706\u9525\u9762\u4e24\u4fa7\u90fd\u76f8\u4ea4\uff0c\u4e14\u4e0d\u8fc7\u5706\u9525\u9876\u70b9\uff0c\u7ed3\u679c\u4e3a\u53cc\u66f2\u7ebf\u7684\u4e00\u652f\uff08\u53e6\u4e00\u652f\u4e3a\u6b64\u5706\u9525\u9762\u7684\u5bf9\u9876\u5706\u9525\u9762\u4e0e\u5e73\u9762\u7684\u4ea4\u7ebf\uff09\u3002
7) \u5f53\u5e73\u9762\u4e0e\u5706\u9525\u9762\u4e24\u4fa7\u90fd\u76f8\u4ea4\uff0c\u4e14\u8fc7\u5706\u9525\u9876\u70b9\uff0c\u7ed3\u679c\u4e3a\u4e24\u6761\u76f8\u4ea4\u76f4\u7ebf\u3002
\u4ee3\u6570\u89c2\u70b9
\u5728\u7b1b\u5361\u5c14\u5e73\u9762\u4e0a\uff0c\u4e8c\u5143\u4e8c\u6b21\u65b9\u7a0bax^2+bxy+cy^2+dx+ey+f=0\u7684\u56fe\u50cf\u662f\u5706\u9525\u66f2\u7ebf\u3002\u6839\u636e\u5224\u522b\u5f0f\u7684\u4e0d\u540c\uff0c\u4e5f\u5305\u542b\u4e86\u692d\u5706\uff0c\u53cc\u66f2\u7ebf\uff0c\u629b\u7269\u7ebf\u4ee5\u53ca\u5404\u79cd\u9000\u5316\u60c5\u5f62\u3002
\u7126\u70b9-\u51c6\u7ebf\u89c2\u70b9
\uff08\u4e25\u683c\u6765\u8bb2\uff0c\u8fd9\u79cd\u89c2\u70b9\u4e0b\u53ea\u80fd\u5b9a\u4e49\u5706\u9525\u66f2\u7ebf\u7684\u51e0\u79cd\u4e3b\u8981\u60c5\u5f62\uff0c\u56e0\u800c\u4e0d\u80fd\u7b97\u662f\u5706\u9525\u66f2\u7ebf\u7684\u5b9a\u4e49\u3002\u4f46\u56e0\u5176\u4f7f\u7528\u5e7f\u6cdb\uff0c\u5e76\u80fd\u5f15\u5bfc\u51fa\u8bb8\u591a\u5706\u9525\u66f2\u7ebf\u4e2d\u91cd\u8981\u7684\u51e0\u4f55\u6982\u5ff5\u548c\u6027\u8d28\uff09\u3002
\u7ed9\u5b9a\u4e00\u70b9P\uff0c\u4e00\u76f4\u7ebfL\u4ee5\u53ca\u4e00\u975e\u8d1f\u5b9e\u5e38\u6570e\uff0c\u5219\u5230P\u7684\u8ddd\u79bb\u4e0eL\u8ddd\u79bb\u4e4b\u6bd4\u4e3ae\u7684\u70b9\u7684\u8f68\u8ff9\u662f\u5706\u9525\u66f2\u7ebf\u3002
\u6839\u636ee\u7684\u8303\u56f4\u4e0d\u540c\uff0c\u66f2\u7ebf\u4e5f\u5404\u4e0d\u76f8\u540c\u3002\u5177\u4f53\u5982\u4e0b\uff1a
1) e=0\uff0c\u8f68\u8ff9\u9000\u5316\u4e3a\u70b9\uff08\u5373\u5b9a\u70b9P\uff09\uff1b
2) e=1\uff08\u5373\u5230P\u4e0e\u5230L\u8ddd\u79bb\u76f8\u540c\uff09\uff0c\u8f68\u8ff9\u4e3a\u629b\u7269\u7ebf\uff1b
3) 0<e<1\uff0c\u8f68\u8ff9\u4e3a\u692d\u5706\uff1b
4) e>1\uff0c\u8f68\u8ff9\u4e3a\u53cc\u66f2\u7ebf\u3002
\u7f16\u8f91\u672c\u6bb5\u6982\u5ff5
\uff08\u4ee5\u4e0b\u4ee5\u7eaf\u51e0\u4f55\u65b9\u5f0f\u53d9\u8ff0\u4e3b\u8981\u7684\u5706\u9525\u66f2\u7ebf\u901a\u7528\u7684\u6982\u5ff5\u548c\u6027\u8d28\uff0c\u7531\u4e8e\u5927\u90e8\u5206\u6027\u8d28\u662f\u5728\u7126\u70b9\uff0d\u51c6\u7ebf\u89c2\u70b9\u4e0b\u5b9a\u4e49\u7684\uff0c\u5bf9\u4e8e\u66f4\u4e00\u822c\u7684\u9000\u5316\u60c5\u5f62\uff0c\u6709\u4e9b\u6982\u5ff5\u53ef\u80fd\u4e0d\u9002\u7528\u3002\uff09
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Pappus\u5b9a\u7406\uff1a\u5706\u9525\u66f2\u7ebf\u4e0a\u4e00\u70b9\u7684\u7126\u534a\u5f84\u957f\u5ea6\u7b49\u4e8e\u8be5\u70b9\u5230\u76f8\u5e94\u51c6\u7ebf\u7684\u8ddd\u79bb\u4e58\u4ee5\u79bb\u5fc3\u7387\u3002
Pascal\u5b9a\u7406\uff1a\u5706\u9525\u66f2\u7ebf\u7684\u5185\u63a5\u516d\u8fb9\u5f62\uff0c\u82e5\u5bf9\u8fb9\u4e24\u4e24\u4e0d\u5e73\u884c\uff0c\u5219\u8be5\u516d\u8fb9\u5f62\u5bf9\u8fb9\u5ef6\u957f\u7ebf\u7684\u4ea4\u70b9\u5171\u7ebf\u3002\uff08\u5bf9\u4e8e\u9000\u5316\u7684\u60c5\u5f62\u4e5f\u9002\u7528\uff09
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PR sina=PH=PE sinb=PF sinb
PF/PR=sina/sinb\u4e3a\u5e38\u6570
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1\uff09\u692d\u5706\uff08Ellipse)
\u6587\u5b57\u8bed\u8a00\u5b9a\u4e49\uff1a\u5e73\u9762\u5185\u4e00\u4e2a\u52a8\u70b9\u5230\u4e00\u4e2a\u5b9a\u70b9\u4e0e\u4e00\u6761\u5b9a\u76f4\u7ebf\u7684\u8ddd\u79bb\u4e4b\u6bd4\u662f\u4e00\u4e2a\u5c0f\u4e8e1\u7684\u6b63\u5e38\u6570e\u3002\u5e73\u9762\u5185\u4e00\u4e2a\u52a8\u70b9\u5230\u4e24\u4e2a\u5b9a\u70b9\uff08\u7126\u70b9\uff09\u7684\u8ddd\u79bb\u548c\u7b49\u4e8e\u5b9a\u957f2a\u7684\u70b9\u7684\u96c6\u5408\u3002\u5b9a\u70b9\u662f\u692d\u5706\u7684\u7126\u70b9\uff0c\u5b9a\u76f4\u7ebf\u662f\u692d\u5706\u7684\u51c6\u7ebf\uff0c\u5e38\u6570e\u662f\u692d\u5706\u7684\u79bb\u5fc3\u7387\u3002
\u6807\u51c6\u65b9\u7a0b\uff1a
1.\u4e2d\u5fc3\u5728\u539f\u70b9\uff0c\u7126\u70b9\u5728x\u8f74\u4e0a\u7684\u692d\u5706\u6807\u51c6\u65b9\u7a0b\uff1a\uff08x^2/a^2)+(y^2/b^2)=1
\u5176\u4e2da>b>0\uff0cc>0\uff0cc^2=a^2-b^2.
2.\u4e2d\u5fc3\u5728\u539f\u70b9\uff0c\u7126\u70b9\u5728y\u8f74\u4e0a\u7684\u692d\u5706\u6807\u51c6\u65b9\u7a0b\uff1a\uff08x^2/b^2)+(y^2/a^2)=1
\u5176\u4e2da>b>0\uff0cc>0\uff0cc^2=a^2-b^2\u3002
\u53c2\u6570\u65b9\u7a0b\uff1ax=acos\u03b8 y=bsin\u03b8 \uff08\u03b8\u4e3a\u53c2\u6570 ,0\u2264\u03b8\u22642\u03c0)
2\uff09\u53cc\u66f2\u7ebf\uff08Hyperbola)
\u6587\u5b57\u8bed\u8a00\u5b9a\u4e49\uff1a\u5e73\u9762\u5185\u4e00\u4e2a\u52a8\u70b9\u5230\u4e00\u4e2a\u5b9a\u70b9\u4e0e\u4e00\u6761\u5b9a\u76f4\u7ebf\u7684\u8ddd\u79bb\u4e4b\u6bd4\u662f\u4e00\u4e2a\u5927\u4e8e1\u7684\u5e38\u6570e\u3002\u5b9a\u70b9\u662f\u53cc\u66f2\u7ebf\u7684\u7126\u70b9\uff0c\u5b9a\u76f4\u7ebf\u662f\u53cc\u66f2\u7ebf\u7684\u51c6\u7ebf\uff0c\u5e38\u6570e\u662f\u53cc\u66f2\u7ebf\u7684\u79bb\u5fc3\u7387\u3002
\u6807\u51c6\u65b9\u7a0b\uff1a
1.\u4e2d\u5fc3\u5728\u539f\u70b9\uff0c\u7126\u70b9\u5728x\u8f74\u4e0a\u7684\u53cc\u66f2\u7ebf\u6807\u51c6\u65b9\u7a0b\uff1a\u3000\uff08x^2/a^2)-(y^2/b^2)=1
\u5176\u4e2da>0,b>0,c^2=a^2+b^2.
2.\u4e2d\u5fc3\u5728\u539f\u70b9\uff0c\u7126\u70b9\u5728y\u8f74\u4e0a\u7684\u53cc\u66f2\u7ebf\u6807\u51c6\u65b9\u7a0b\uff1a\u3000\uff08y^2/a^2)-(x^2/b^2)=1.
\u5176\u4e2da>0,b>0,c^2=a^2+b^2.
\u53c2\u6570\u65b9\u7a0b\uff1ax=asec\u03b8 y=btan\u03b8 \uff08\u03b8\u4e3a\u53c2\u6570 )
\u76f4\u89d2\u5750\u6807\uff08\u4e2d\u5fc3\u4e3a\u539f\u70b9\uff09\uff1ax^2/a^2 - y^2/b^2 = 1 \uff08\u5f00\u53e3\u65b9\u5411\u4e3ax\u8f74\uff09 y^2/a^2 - x^2/b^2 = 1 \uff08\u5f00\u53e3\u65b9\u5411\u4e3ay\u8f74\uff09
3\uff09\u629b\u7269\u7ebf\uff08Parabola)
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\u53c2\u6570\u65b9\u7a0b
x=2pt^2 y=2pt (t\u4e3a\u53c2\u6570\uff09 t=1/tan\u03b8\uff08tan\u03b8\u4e3a\u66f2\u7ebf\u4e0a\u70b9\u4e0e\u5750\u6807\u539f\u70b9\u786e\u5b9a\u76f4\u7ebf\u7684\u659c\u7387\uff09\u7279\u522b\u5730\uff0ct\u53ef\u7b49\u4e8e0
\u76f4\u89d2\u5750\u6807
y=ax^2+bx+c \uff08\u5f00\u53e3\u65b9\u5411\u4e3ay\u8f74\uff0ca\u22600\uff09 x=ay^2+by+c \uff08\u5f00\u53e3\u65b9\u5411\u4e3ax\u8f74\uff0ca\u22600 )
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\u692d\u5706
|PF1|=a+ex
|PF2|=a-ex
\u53cc\u66f2\u7ebf
P\u5728\u5de6\u652f\uff0c|PF1|=\uff0da-ex |PF2|=a-ex
P\u5728\u53f3\u652f\uff0c|PF1|=a+ex |PF2|=\uff0da+ex
P\u5728\u4e0b\u652f\uff0c|PF1|= \uff0da-ey |PF2|=a-ey
P\u5728\u4e0a\u652f\uff0c|PF1|= a+ey |PF2|=\uff0da+ey
\u629b\u7269\u7ebf
|PF|=x+p/2
\u5706\u9525\u66f2\u7ebf\u7684\u5207\u7ebf\u65b9\u7a0b
\u5706\u9525\u66f2\u7ebf\u4e0a\u4e00\u70b9P\uff08x0,y0\uff09\u7684\u5207\u7ebf\u65b9\u7a0b\u4ee5x0x\u4ee3\u66ffx^2\uff0c\u4ee5y0y\u4ee3\u66ffy^2\uff1b\u4ee5\uff08x0+x)/2\u4ee3\u66ffx\uff0c\u4ee5\uff08y0+y)/2\u4ee3\u66ffy
\u5373\u692d\u5706\uff1ax0x/a^2+y0y/b^2=1\uff1b\u53cc\u66f2\u7ebf\uff1ax0x/a^2-y0y/b^2=1\uff1b\u629b\u7269\u7ebf\uff1ay0y=p(x0+x)
\u7126\u51c6\u8ddd
\u5706\u9525\u66f2\u7ebf\u7684\u7126\u70b9\u5230\u51c6\u7ebf\u7684\u8ddd\u79bbp\u53eb\u5706\u9525\u66f2\u7ebf\u7684\u7126\u51c6\u8ddd\uff0c\u6216\u7126\u53c2\u6570\u3002
\u692d\u5706\u7684\u7126\u51c6\u8ddd\uff1ap=(b^2)/c
\u53cc\u66f2\u7ebf\u7684\u7126\u51c6\u8ddd\uff1ap=(b^2)/c
\u629b\u7269\u7ebf\u7684\u51c6\u7126\u8ddd\uff1ap
\u7126\u70b9\u4e09\u89d2\u5f62
\u692d\u5706\u6216\u53cc\u66f2\u7ebf\u4e0a\u7684\u4e00\u70b9\u4e0e\u4e24\u7126\u70b9\u6240\u6784\u6210\u7684\u4e09\u89d2\u5f62\u3002
\u8bbeF1 F2\u5206\u522b\u4e3a\u692d\u5706\u6216\u53cc\u66f2\u7ebf\u7684\u4e24\u4e2a\u7126\u70b9\uff0cP\u4e3a\u692d\u5706\u6216\u53cc\u66f2\u7ebf\u4e0a\u7684\u4e00\u70b9\u4e14PF1F2\u80fd\u6784\u6210\u4e09\u89d2\u5f62\u3002
\u82e5\u2220F1PF2=\u03b8\uff0c\u5219\u692d\u5706\u7126\u70b9\u4e09\u89d2\u5f62\u7684\u9762\u79ef\u4e3aS=b^2tan\uff08\u03b8/2\uff09.\u53cc\u66f2\u7ebf\u7126\u70b9\u4e09\u89d2\u5f62\u7684\u9762\u79ef\u4e3aS=b^2cot\uff08\u03b8/2\uff09\u3002
\u901a\u5f84
\u5706\u9525\u66f2\u7ebf\u4e2d\uff0c\u8fc7\u7126\u70b9\u5e76\u5782\u76f4\u4e8e\u8f74\u7684\u5f26\u79f0\u4e3a\u901a\u5f84\u3002
\u692d\u5706\u7684\u901a\u5f84\uff1a\uff082b^2)/a
\u53cc\u66f2\u7ebf\u7684\u901a\u5f84\uff1a\uff082b^2)/a
\u629b\u7269\u7ebf\u7684\u901a\u5f84\uff1a2p
\u5706\u9525\u66f2\u7ebf\u7684\u6027\u8d28\u5bf9\u6bd4
\u5706\u9525\u66f2\u7ebf
\u692d\u5706
\u53cc\u66f2\u7ebf
\u629b\u7269\u7ebf
\u6807\u51c6\u65b9\u7a0b
(x^2/a^2)+(y^2/b^2)=1
a>b>0
(x^2/a^2)-(y^2/b^2)=1
a>0,b>0
y^2=2px
p>0
\u8303\u56f4
x\u2208[-a,a]
y\u2208[-b,b]
x\u2208\uff08-\u221e\uff0c-a]\u222a[a,+\u221e\uff09
y\u2208R
x\u2208[0,+\u221e\uff09
y\u2208R
\u5bf9\u79f0\u6027
\u5173\u4e8ex\u8f74\uff0cy\u8f74\uff0c\u539f\u70b9\u5bf9\u79f0
\u5173\u4e8ex\u8f74\uff0cy\u8f74\uff0c\u539f\u70b9\u5bf9\u79f0
\u5173\u4e8ex\u8f74\u5bf9\u79f0
\u9876\u70b9
(a,0),(-a,0),(0,b),(0,-b)
(a,0),(-a,0)
(0,0)
\u7126\u70b9
(c,0),(-c,0)
\u3010\u5176\u4e2dc^2=a^2-b^2\u3011
(c,0),(-c,0)
\u3010\u5176\u4e2dc^2=a^2+b^2\u3011
(p/2,0)
\u51c6\u7ebf
x=\u00b1\uff08a^2)/c
x=\u00b1\uff08a^2)/c
x=-p/2
\u6e10\u8fd1\u7ebf
\u2014\u2014\u2014\u2014\u2014\u2014
y=\u00b1(b/a)x[1]
\u2014\u2014\u2014\u2014\u2014
\u79bb\u5fc3\u7387
e=c/a,e\u2208\uff080,1)
e=c/a,e\u2208\uff081,+\u221e\uff09
e=1
\u7126\u534a\u5f84
\u2223PF1\u2223=a+ex
\u2223PF2\u2223=a-ex
\u2223PF1\u2223=\u2223ex+a\u2223
\u2223PF2\u2223=\u2223ex-a\u2223
\u2223PF\u2223=x+p/2
\u7126\u51c6\u8ddd
p=(b^2)/c
p=(b^2)/c
p
\u901a\u5f84
(2b^2)/a
(2b^2)/a
2p
\u53c2\u6570\u65b9\u7a0b
x=a\u00b7cos\u03b8
y=b\u00b7sin\u03b8\uff0c\u03b8\u4e3a\u53c2\u6570
x=a\u00b7sec\u03b8
y=b\u00b7tan\u03b8\uff0c\u03b8\u4e3a\u53c2\u6570
x=2pt^2
y=2pt,t\u4e3a\u53c2\u6570
\u8fc7\u5706\u9525\u66f2\u7ebf\u4e0a\u4e00\u70b9
(x0,y0\uff09\u7684\u5207\u7ebf\u65b9\u7a0b
(x0\u00b7x/a^2)+(y0\u00b7y/b^2)=1
(x0x/a^2)-(y0\u00b7y/b^2)=1
y0\u00b7y=p(x+x0)
\u659c\u7387\u4e3ak\u7684\u5207\u7ebf\u65b9\u7a0b
y=kx\u00b1\u221a[(a^2\uff09\u00b7\uff08k^2)+b^2]
y=kx\u00b1\u221a[(a^2\uff09\u00b7\uff08k^2)-b^2]
y=kx+p/2k
\u5706\u9525\u66f2\u7ebf\u7684\u4e2d\u70b9\u5f26\u95ee\u9898
\u5df2\u77e5\u5706\u9525\u66f2\u7ebf\u5185\u4e00\u70b9\u4e3a\u5706\u9525\u66f2\u7ebf\u7684\u4e00\u5f26\u4e2d\u70b9\uff0c\u6c42\u8be5\u5f26\u7684\u65b9\u7a0b\uff1a
\u2488\u8054\u7acb\u65b9\u7a0b\u6cd5\u3002
\u7528\u70b9\u659c\u5f0f\u8bbe\u51fa\u8be5\u5f26\u7684\u65b9\u7a0b\uff08\u659c\u7387\u4e0d\u5b58\u5728\u7684\u60c5\u51b5\u9700\u8981\u53e6\u5916\u8003\u8651\uff09\uff0c\u4e0e\u5706\u9525\u66f2\u7ebf\u65b9\u7a0b\u8054\u7acb\u6c42\u5f97\u5173\u4e8ex\u7684\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\u548c\u5173\u4e8ey\u7684\u4e00\u5143\u4e8c\u6b21\u65b9\u7a0b\uff0c\u7531\u97e6\u8fbe\u5b9a\u7406\u5f97\u5230\u4e24\u6839\u4e4b\u548c\u7684\u8868\u8fbe\u5f0f\uff0c\u5728\u7531\u4e2d\u70b9\u5750\u6807\u516c\u5f0f\u7684\u4e24\u6839\u4e4b\u548c\u7684\u5177\u4f53\u6570\u503c\uff0c\u6c42\u51fa\u8be5\u5f26\u7684\u65b9\u7a0b\u3002
2.\u70b9\u5dee\u6cd5\uff0c\u6216\u79f0\u4ee3\u70b9\u76f8\u51cf\u6cd5\u3002
\u8bbe\u51fa\u5f26\u7684\u4e24\u7aef\u70b9\u5750\u6807\uff08x1,y1\uff09\u548c\uff08x2,y2\uff09\uff0c\u4ee3\u5165\u5706\u9525\u66f2\u7ebf\u7684\u65b9\u7a0b\uff0c\u5c06\u5f97\u5230\u7684\u4e24\u4e2a\u65b9\u7a0b\u76f8\u51cf\uff0c\u8fd0\u7528\u5e73\u65b9\u5dee\u516c\u5f0f\u5f97[(x1+x2\uff09\u00b7\uff08x1-x2)]/(a^2)+[(y1+y2\uff09\u00b7\uff08y1-y2)/(b^2]=0
\u7531\u659c\u7387\u4e3a\uff08y1-y2)/(x1-x2\uff09\u53ef\u4ee5\u5f97\u5230\u659c\u7387\u7684\u53d6\u503c\u3002\uff08\u4f7f\u7528\u65f6\u6ce8\u610f\u5224\u522b\u5f0f\u7684\u95ee\u9898\uff09
\u6c42\u70b9\u7684\u8f68\u8ff9\u65b9\u7a0b
\u5728\u6c42\u66f2\u7ebf\u7684\u8f68\u8ff9\u65b9\u7a0b\u65f6\uff0c\u5982\u679c\u80fd\u591f\u5c06\u9898\u8bbe\u6761\u4ef6\u8f6c\u5316\u4e3a\u5177\u6709\u67d0\u79cd\u52a8\u611f\u7684\u76f4\u89c2\u56fe\u5f62\uff0c\u901a\u8fc7\u89c2\u5bdf\u56fe\u5f62\u7684\u53d8\u5316\u8fc7\u7a0b\uff0c\u53d1\u73b0\u5176\u5185\u5728\u8054\u7cfb\uff0c\u627e\u51fa\u54ea\u4e9b\u662f\u53d8\u5316\u7684\u91cf\uff08\u6216\u5173\u7cfb\uff09\u3001\u54ea\u4e9b\u662f\u59cb\u7ec8\u4fdd\u6301\u4e0d\u53d8\u7684\u91cf\uff08\u6216\u5173\u7cfb\uff09\uff0c\u90a3\u4e48\u6211\u4eec\u5c31\u53ef\u4ee5\u4ece\u627e\u51fa\u7684\u4e0d\u53d8\u91cf\uff08\u6216\u5173\u7cfb\uff09\u51fa\u53d1\uff0c\u6253\u5f00\u89e3\u9898\u601d\u8def\uff0c\u786e\u5b9a\u89e3\u9898\u65b9\u6cd5\u3002
\u5706\u9525\u66f2\u7ebf\u7684\u66f2\u7387\uff08\u89c1\u53f3\u56fe\uff09\u66f2\u7387\u534a\u5f84\u7684\u4f5c\u56fe\u3002\u7b2c\u4e8c\u6761\u5782\u7ebf\u4e0e\u6cd5\u7ebf\u7684\u4ea4\u70b9
Z\u5c31\u662f\u66f2\u7387\u7684\u4e2d\u5fc3\u5b83\u5230P\u70b9\u7684\u8ddd\u79bb\u4fbf\u662f\u66f2\u7387\u534a\u5f84\u3002
\u7f16\u8f91\u672c\u6bb5\u5224\u522b\u6cd5
\u8bbe\u5706\u9525\u66f2\u7ebf\u7684\u65b9\u7a0b\u4e3a
Ax^2+2Bxy+Cy^2+2Dx+2Ey+F=0
|A B D|
?= |B C E| \u03b4=|A B| S=A+C \u79f0\u4e3a\u4e8c\u6b21\u66f2\u7ebf\u4e0d\u53d8\u91cf
|D E F|\u3000 |B C|
\u03b4>0
=0
\u6709\u4e00\u5b9e\u70b9\u7684\u76f8\u4ea4\u865a\u76f4\u7ebf
\u03b4>0
\u22600
S<0
\u692d\u5706
\u03b4>0
\u22600
S>0
\u865a\u692d\u5706
\u03b4<0
=0
\u76f8\u4ea4\u76f4\u7ebf
\u03b4<0
\u22600
\u53cc\u66f2\u7ebf
\u03b4=0
\u22600
\u629b\u7269\u7ebf
\u03b4=0
=0
D^2+E^2-AF-CF>0
\u5e73\u884c\u76f4\u7ebf
\u03b4=0
=0
D^2+E^2-AF-CF=0
\u91cd\u5408\u76f4\u7ebf
\u03b4=0
=0
D^2+E^2-AF-CF<0
\u5e73\u884c\u865a\u76f4\u7ebf
\u7f16\u8f91\u672c\u6bb5\u6f2b\u8c08
\u5706\u9525\u66f2\u7ebf\u5305\u62ec\u692d\u5706\u3001\u629b\u7269\u7ebf\u3001\u53cc\u66f2\u7ebf\u548c\u5706\uff0c\u901a\u8fc7\u76f4\u89d2\u5750\u6807\u7cfb\uff0c\u5b83\u4eec\u53c8\u4e0e\u4e8c\u6b21\u65b9\u7a0b\u5bf9\u5e94\uff0c\u6240\u4ee5\uff0c\u5706\u9525\u66f2\u7ebf\u53c8\u53eb\u505a\u4e8c\u6b21\u66f2\u7ebf\u3002\u5706\u9525\u66f2\u7ebf\u4e00\u76f4\u662f\u51e0\u4f55\u5b66\u7814\u7a76\u7684\u91cd\u8981\u8bfe\u9898\u4e4b\u4e00\uff0c\u5728\u6211\u4eec\u7684\u5b9e\u9645\u751f\u6d3b\u4e2d\u4e5f\u5b58\u5728\u7740\u8bb8\u8bb8\u591a\u591a\u7684\u5706\u9525\u66f2\u7ebf\u3002
\u6211\u4eec\u751f\u6d3b\u7684\u5730\u7403\u6bcf\u65f6\u6bcf\u523b\u90fd\u5728\u73af\u7ed5\u592a\u9633\u7684\u692d\u5706\u8f68\u8ff9\u4e0a\u8fd0\u884c\uff0c\u592a\u9633\u7cfb\u5176\u4ed6\u884c\u661f\u4e5f\u5982\u6b64\uff0c\u592a\u9633\u5219\u4f4d\u4e8e\u692d\u5706\u7684\u4e00\u4e2a\u7126\u70b9\u4e0a\u3002\u5982\u679c\u8fd9\u4e9b\u884c\u661f\u8fd0\u884c\u901f\u5ea6\u589e\u5927\u5230\u67d0\u79cd\u7a0b\u5ea6\uff0c\u5b83\u4eec\u5c31\u4f1a\u6cbf\u629b\u7269\u7ebf\u6216\u53cc\u66f2\u7ebf\u8fd0\u884c\u3002\u4eba\u7c7b\u53d1\u5c04\u4eba\u9020\u5730\u7403\u536b\u661f\u6216\u4eba\u9020\u884c\u661f\u5c31\u8981\u9075\u7167\u8fd9\u4e2a\u539f\u7406\u3002\u76f8\u5bf9\u4e8e\u4e00\u4e2a\u7269\u4f53\uff0c\u6309\u4e07\u6709\u5f15\u529b\u5b9a\u5f8b\u53d7\u5b83\u5438\u5f15\u7684\u53e6\u4e00\u7269\u4f53\u7684\u8fd0\u52a8\uff0c\u4e0d\u53ef\u80fd\u6709\u4efb\u4f55\u5176\u4ed6\u7684\u8f68\u9053\u4e86\u3002\u56e0\u800c\uff0c\u5706\u9525\u66f2\u7ebf\u5728\u8fd9\u79cd\u610f\u4e49\u4e0a\u8bb2\uff0c\u5b83\u6784\u6210\u4e86\u6211\u4eec\u5b87\u5b99\u7684\u57fa\u672c\u5f62\u5f0f\u3002
\u7531\u629b\u7269\u7ebf\u7ed5\u5176\u8f74\u65cb\u8f6c\uff0c\u53ef\u5f97\u5230\u4e00\u4e2a\u53eb\u505a\u65cb\u8f6c\u7269\u9762\u7684\u66f2\u9762\u3002\u5b83\u4e5f\u6709\u4e00\u6761\u8f74\uff0c\u5373\u629b\u7269\u7ebf\u7684\u8f74\u3002\u5728\u8fd9\u4e2a\u8f74\u4e0a\u6709\u4e00\u4e2a\u5177\u6709\u5947\u5999\u6027\u8d28\u7684\u7126\u70b9\uff0c\u4efb\u4f55\u4e00\u6761\u8fc7\u7126\u70b9\u7684\u76f4\u7ebf\u7531\u629b\u7269\u9762\u53cd\u5c04\u51fa\u6765\u4ee5\u540e\uff0c\u90fd\u6210\u4e3a\u5e73\u884c\u4e8e\u8f74\u7684\u76f4\u7ebf\u3002\u8fd9\u5c31\u662f\u6211\u4eec\u4e3a\u4ec0\u4e48\u8981\u628a\u63a2\u7167\u706f\u53cd\u5149\u955c\u505a\u6210\u65cb\u8f6c\u629b\u7269\u9762\u7684\u9053\u7406\u3002
\u7531\u53cc\u66f2\u7ebf\u7ed5\u5176\u865a\u8f74\u65cb\u8f6c\uff0c\u53ef\u4ee5\u5f97\u5230\u5355\u53f6\u53cc\u66f2\u9762\uff0c\u5b83\u53c8\u662f\u4e00\u79cd\u76f4\u7eb9\u66f2\u9762\uff0c\u7531\u4e24\u7ec4\u6bcd\u76f4\u7ebf\u65cf\u7ec4\u6210\uff0c\u5404\u7ec4\u5185\u6bcd\u76f4\u7ebf\u4e92\u4e0d\u76f8\u4ea4\uff0c\u800c\u4e0e\u53e6\u4e00\u7ec4\u6bcd\u76f4\u7ebf\u5374\u76f8\u4ea4\u3002\u4eba\u4eec\u5728\u8bbe\u8ba1\u9ad8\u5927\u7684\u7acb\u5854\uff08\u5982\u51b7\u5374\u5854\uff09\u65f6\uff0c\u5c31\u91c7\u53d6\u5355\u53f6\u53cc\u66f2\u9762\u7684\u4f53\u5f62\uff0c\u65e2\u8f7b\u5de7\u53c8\u575a\u56fa\u3002
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\u7f16\u8f91\u672c\u6bb5\u7814\u7a76\u5386\u53f2
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1. 椭圆:到两个定点的距离之和等于定长(定长大于两个定点间的距离)的动点的轨迹叫做椭圆。即:{P| |PF<sub>1</sub>|+|PF<sub>2</sub>|=2a, (2a>|F<sub>1</sub>F<sub>2</sub>|)}。
2. 双曲线:到两个定点的距离的差的绝对值为定值(定值小于两个定点的距离)的动点轨迹叫做双曲线。即{P|||PF<sub>1</sub>|-|PF<sub>2</sub>||=2a, (2a<|F<sub>1</sub>F<sub>2</sub>|)}。
3. 圆锥曲线的统一定义:到定点的距离与到定直线的距离的比e是常数的点的轨迹叫做圆锥曲线。当0时为椭圆:当e=1时为抛物线;当e>1时为双曲线。
二、圆锥曲线的方程。
1.椭圆: + =1(a>b>0)或 + =1(a>b>0)(其中,a2=b2+c2)
2.双曲线: - =1(a>0, b>0)或 - =1(a>0, b>0)(其中,c2=a2+b2)
3.抛物线:y2=±2px(p>0),x2=±2py(p>0)
三、圆锥曲线的性质
1.椭圆: + =1(a>b>0)
(1)范围:|x|≤a,|y|≤b
(2)顶点:(±a,0),(0,±b)
(3)焦点:(±c,0)
(4)离心率:e= ∈(0,1)
(5)准线:x=±
2.双曲线: - =1(a>0, b>0)
(1)范围:|x|≥a, y∈R
(2)顶点:(±a,0)
(3)焦点:(±c,0)
(4)离心率:e= ∈(1,+∞)
(5)准线:x=±
(6)渐近线:y=± x
3.抛物线:y2=2px(p>0)
(1)范围:x≥0, y∈R
(2)顶点:(0,0)
(3)焦点:( ,0)
(4)离心率:e=1
(5)准线:x=- 四、例题选讲: 例1.椭圆短轴长为2,长轴是短轴的2倍,则椭圆中心到准线的距离是__________。
解:由题:2b=2,b=1,a=2,c= = ,则椭圆中心到准线的距离: = = 。
注意:椭圆本身的性质(如焦距,中心到准线的距离,焦点到准线的距离等等)不受椭圆的位置的影响。
例2.椭圆 + =1的离心率e= ,则m=___________。
解:(1)椭圆的焦点在x轴上,a2=m,b2=4,c2=m-4,e2= = = m=8。
(2)椭圆的焦点在y轴上,a2=4,b2=m,c2=4-m,e2= = = m=2。
注意:椭圆方程的标准形式有两个,在没有确定的情况下,两种情况都要考虑,切不可凭主观丢掉一解。
例3.如图:椭圆 + =1(a>b>0),F1为左焦点,A、B是两个顶点,P为椭圆上一点,PF1⊥x轴,且PO//AB,求椭圆的离心率e。
解:设椭圆的右焦点为F2,由第一定义:|PF1|+|PF2|=2a,
∵ PF1⊥x轴,∴ |PF1|2+|F1F2|2=|PF2|2,
即(|PF2|+|PF1|)(|PF2|-|PF1|)=4c2, ∴ |PF1|= 。 ∵ PO//AB,∴ ΔPF1O∽ΔBOA,
∴ = c=b a= c, ∴ e= = 。
又解,∵ PF1⊥x轴,∴ 设P(-c, y)。
由第二定义: =e |PF1|=e(x0+ )= (-c+ )= ,
由上解中ΔPF1O∽ΔBOA,得到b=c e= 。
例4.已知F1,F2为椭圆 + =1的焦点,P为椭圆上一点,且∠F1PF2= ,求ΔF1PF2的面积。
分析:要求三角形的面积,可以直接利用三角形的面积公式,注意到椭圆中一些量之间的关系,我们选用面积公式S= absinC。
解法 一:SΔ= |PF1|·|PF2|·sin
|PF1|+|PF2|=2a=20,
4×36=4c2=|F1F2|2=|PF1|2+|PF2|2-2|PF1||PF2|cos ,
即(|PF1|+|PF2|)2-3|PF1||PF2|=4×36,
|PF1|·|PF2|= ∴ SΔ= × × = 。 解法二:SΔ= |F1F2|·|yP|= ×12×yP=6|yP|, 由第二定义: =e |PF1|=a+exP=10+ xP, 由第一定义:|PF2|=2a-|PF1|=10- xP, 4c2=|F1F2|2=(10+ xP)2+(10- xP)2-2(10+ xP)(10- xP)cos , 144=100+ = , =64(1- )=64× , SΔ=6|yP|=6× = 。 注意:两个定义联合运用解决问题。从三角形面积公式均可得到结果。初学时最好两种
圆锥曲线统一定义:(第二定义)
平面上到定点(焦点)的距离与到定直线(准线)的距离为定值(离心率e)的点的集合。而根据e的大小分为椭圆,抛物线,双曲线。圆可看作e为0的曲线。
1。0<e<1为椭圆,直角坐标系中标准方程为:
x^2/a^2+y^2/b^2=1(0<b<a),焦点在x轴上,焦点(c,0)(-c,0)准线x=+-a^2/c,e=c/a
y^2/a^2+y^2/b^2=1(0<b<a),焦点在y轴上,焦点(0,c)(0。-c)准线y=+-a^2/c,e=c/a
a^2=b^2+c^2
椭圆上任意一点到两焦点距离之和为2a(定值),且大于焦距2c,这是第一定义
光学性质:过焦点的任意一条光线经椭圆反射必过另一焦点
2。e=1为抛物线,直角坐标系中标准方程为:
y^2=2px,对称轴为x轴,焦点(p/2,0),准线x=-p/2
x^2=2py,对称轴为y轴,焦点,(0,p/2)准线y=-p/2
光学性质:任意平行对称轴的光线经抛物线反射必过焦点(或反向延长线过焦点)
3。1<e为双曲线,直角坐标系中标准方程为:
x^2/a^2-y^2/b^2=1(0<b<a),焦点在x轴上,焦点(c,0)(-c,0)准线x=+-a^2/c,e=c/a
y^2/a^2-y^2/b^2=1(0<b<a),焦点在y轴上,焦点(0,c)(0。-c)准线y=+-a^2/c,e=c/a
c^2=b^2+a^2
双曲线上任意一点到两焦点距离之差的绝对值为2a(定值),且小于焦距2c,这是第一定义
光学性质:过焦点的任意一条光线经双曲线反射其反向延长线必过另一焦点
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