双曲线 椭圆 抛物线 公式 双曲线'椭圆'抛物线的焦点坐标分别怎么求？公式是什么？
\u53cc\u66f2\u7ebf\uff0c\u692d\u5706\uff0c\u629b\u7269\u7ebf\u7684\u57fa\u672c\u516c\u5f0f\u53cc\u66f2\u7ebf\u7684\u6807\u51c6\u516c\u5f0f\u4e3a\uff1a\u3000\u3000X^2/a^2 - Y^2/b^2 = 1(a>0,b>0)
\u3000\u3000\u800c\u53cd\u6bd4\u4f8b\u51fd\u6570\u7684\u6807\u51c6\u578b\u662f xy = c (c \u2260 0)
\u3000\u3000\u4f46\u662f\u53cd\u6bd4\u4f8b\u51fd\u6570\u786e\u5b9e\u662f\u53cc\u66f2\u7ebf\u51fd\u6570\u7ecf\u8fc7\u65cb\u8f6c\u5f97\u5230\u7684
\u3000\u3000\u56e0\u4e3axy = c\u7684\u5bf9\u79f0\u8f74\u662f y=x, y=-x \u800cX^2/a^2 - Y^2/b^2 = 1\u7684\u5bf9\u79f0\u8f74\u662fx\u8f74\uff0cy\u8f74
\u3000\u3000\u6240\u4ee5\u5e94\u8be5\u65cb\u8f6c45\u5ea6
\u3000\u3000\u8bbe\u65cb\u8f6c\u7684\u89d2\u5ea6\u4e3a a \uff08a\u22600,\u987a\u65f6\u9488\uff09
\u3000\u3000(a\u4e3a\u53cc\u66f2\u7ebf\u6e10\u8fdb\u7ebf\u7684\u503e\u659c\u89d2)
\u3000\u3000\u5219\u6709
\u3000\u3000X = xcosa + ysina
\u3000\u3000Y = - xsina + ycosa
\u3000\u3000\u53d6 a = \u03c0/4
\u3000\u3000\u5219
\u3000\u3000X^2 - Y^2 = (xcos(\u03c0/4) + ysin(\u03c0/4))^2 -(xsin(\u03c0/4) - ycos(\u03c0/4))^2
\u3000\u3000= (\u221a2/2 x + \u221a2/2 y)^2 -(\u221a2/2 x - \u221a2/2 y)^2
\u3000\u3000= 4 (\u221a2/2 x) (\u221a2/2 y)
\u3000\u3000= 2xy.
\u3000\u3000\u800cxy=c
\u3000\u3000\u6240\u4ee5
\u3000\u3000X^2/(2c) - Y^2/(2c) = 1 (c>0)
\u3000\u3000Y^2/(-2c) - X^2/(-2c) = 1 (c<0)
\u3000\u3000\u7531\u6b64\u8bc1\u5f97\uff0c\u53cd\u6bd4\u4f8b\u51fd\u6570\u5176\u5b9e\u5c31\u662f\u53cc\u66f2\u7ebf\u51fd\u6570 \u692d\u5706\u7684\u9762\u79ef\u516c\u5f0f
\u3000\u3000S=\u03c0(\u5706\u5468\u7387)\u00d7a\u00d7b(\u5176\u4e2da,b\u5206\u522b\u662f\u692d\u5706\u7684\u957f\u534a\u8f74,\u77ed\u534a\u8f74\u7684\u957f).
\u3000\u3000\u6216S=\u03c0(\u5706\u5468\u7387)\u00d7A\u00d7B/4(\u5176\u4e2dA,B\u5206\u522b\u662f\u692d\u5706\u7684\u957f\u8f74,\u77ed\u8f74\u7684\u957f).
\u3000\u3000\u692d\u5706\u7684\u5468\u957f\u516c\u5f0f
\u3000\u3000\u692d\u5706\u5468\u957f\u6ca1\u6709\u516c\u5f0f\uff0c\u6709\u79ef\u5206\u5f0f\u6216\u65e0\u9650\u9879\u5c55\u5f00\u5f0f\u3002
\u3000\u3000\u692d\u5706\u5468\u957f(L)\u7684\u7cbe\u786e\u8ba1\u7b97\u8981\u7528\u5230\u79ef\u5206\u6216\u65e0\u7a77\u7ea7\u6570\u7684\u6c42\u548c\u3002\u5982
\u3000\u3000L = \u222b[0,\u03c0/2]4a * sqrt(1-(e*cost)^2)dt\u22482\u03c0\u221a((a^2+b^2)/2) [\u692d\u5706\u8fd1\u4f3c\u5468\u957f], \u5176\u4e2da\u4e3a\u692d\u5706\u957f\u534a\u8f74,e\u4e3a\u79bb\u5fc3\u7387
\u3000\u3000\u692d\u5706\u79bb\u5fc3\u7387\u7684\u5b9a\u4e49\u4e3a\u692d\u5706\u4e0a\u7684\u70b9\u5230\u67d0\u7126\u70b9\u7684\u8ddd\u79bb\u548c\u8be5\u70b9\u5230\u8be5\u7126\u70b9\u5bf9\u5e94\u7684\u51c6\u7ebf\u7684\u8ddd\u79bb\u4e4b\u6bd4\uff0c\u8bbe\u692d\u5706\u4e0a\u70b9P\u5230\u67d0\u7126\u70b9\u8ddd\u79bb\u4e3aPF\uff0c\u5230\u5bf9\u5e94\u51c6\u7ebf\u8ddd\u79bb\u4e3aPL\uff0c\u5219
\u3000\u3000e=PF/PL
\u3000\u3000\u692d\u5706\u7684\u51c6\u7ebf\u65b9\u7a0b
\u3000\u3000x=\u00b1a^2/C
\u3000\u3000\u692d\u5706\u7684\u79bb\u5fc3\u7387\u516c\u5f0f
\u3000\u3000e=c/a(e2c)
\u3000\u3000\u692d\u5706\u7684\u7126\u51c6\u8ddd \uff1a\u692d\u5706\u7684\u7126\u70b9\u4e0e\u5176\u76f8\u5e94\u51c6\u7ebf(\u5982\u7126\u70b9\uff08c,0\uff09\u4e0e\u51c6\u7ebfx=+a^2/C)\u7684\u8ddd\u79bb,\u6570\u503c=b^2/c
\u3000\u3000\u692d\u5706\u7126\u534a\u5f84\u516c\u5f0f \uff5cPF1\uff5c=a+ex0 \uff5cPF2\uff5c=a-ex0
\u3000\u3000\u692d\u5706\u8fc7\u53f3\u7126\u70b9\u7684\u534a\u5f84r=a-ex
\u3000\u3000\u8fc7\u5de6\u7126\u70b9\u7684\u534a\u5f84r=a+ex
\u3000\u3000\u692d\u5706\u7684\u901a\u5f84\uff1a\u8fc7\u7126\u70b9\u7684\u5782\u76f4\u4e8ex\u8f74\uff08\u6216y\u8f74\uff09\u7684\u76f4\u7ebf\u4e0e\u692d\u5706\u7684\u4e24\u7126\u70b9A,B\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\u6570\u503c=2b^2/a
\u3000\u3000\u70b9\u4e0e\u692d\u5706\u4f4d\u7f6e\u5173\u7cfb \u70b9M\uff08x0\uff0cy0\uff09 \u692d\u5706 x^2/a^2+y^2/b^2=1
\u3000\u3000\u70b9\u5728\u5706\u5185\uff1a x0^2/a^2+y0^2/b^2\uff1c1
\u3000\u3000\u70b9\u5728\u5706\u4e0a\uff1a x0^2/a^2+y0^2/b^2=1
\u3000\u3000\u70b9\u5728\u5706\u5916\uff1a x0^2/a^2+y0^2/b^2\uff1e1
\u3000\u3000\u76f4\u7ebf\u4e0e\u692d\u5706\u4f4d\u7f6e\u5173\u7cfb
\u3000\u3000y=kx+m \u2460
\u3000\u3000x^2/a^2+y^2/b^2=1 \u2461
\u3000\u3000\u7531\u2460\u2461\u53ef\u63a8\u51fax^2/a^2+\uff08kx+m\uff09^2/b^2=1
\u3000\u3000\u76f8\u5207\u25b3=0
\u3000\u3000\u76f8\u79bb\u25b3\uff1c0\u65e0\u4ea4\u70b9
\u3000\u3000\u76f8\u4ea4\u25b3\uff1e0 \u53ef\u5229\u7528\u5f26\u957f\u516c\u5f0f\uff1aA(x1,y1) B(x2,y2)
\u3000\u3000|AB|=d = \u221a(1+k^2)|x1-x2| = \u221a(1+k^2)(x1-x2)^2 = \u221a(1+1/k^2)|y1-y2| = \u221a(1+1/k^2)(y1-y2)^2
\u3000\u3000\u692d\u5706\u901a\u5f84\uff08\u5b9a\u4e49\uff1a\u5706\u9525\u66f2\u7ebf\uff08\u9664\u5706\u5916\uff09\u4e2d\uff0c\u8fc7\u7126\u70b9\u5e76\u5782\u76f4\u4e8e\u8f74\u7684\u5f26\uff09\u516c\u5f0f\uff1a2b^2/a
\u3000\u3000\u692d\u5706\u7684\u659c\u7387\u516c\u5f0f\u3000\u8fc7\u692d\u5706\u4e0ax^2/a^2+y^2/b^2\u4e0a\u4e00\u70b9\uff08x\uff0cy\uff09\u7684\u5207\u7ebf\u659c\u7387\u4e3ab^2*X/a^2y \u629b\u7269\u7ebf\u7684\u6807\u51c6\u65b9\u7a0b\u3000\u3000\u53f3\u5f00\u53e3\u629b\u7269\u7ebf:y^2=2px
\u3000\u3000\u5de6\u5f00\u53e3\u629b\u7269\u7ebf:y^2=-2px
\u3000\u3000\u4e0a\u5f00\u53e3\u629b\u7269\u7ebf:x^2=2py
\u3000\u3000\u4e0b\u5f00\u53e3\u629b\u7269\u7ebf:x^2=-2py
\u3000\u3000p\u4e3a\u7126\u51c6\u8ddd\uff08p>0\uff09 [\u7f16\u8f91\u672c\u6bb5]3.\u629b\u7269\u7ebf\u76f8\u5173\u53c2\u6570(\u5bf9\u4e8e\u5411\u53f3\u5f00\u53e3\u7684\u629b\u7269\u7ebf) \u3000\u3000\u79bb\u5fc3\u7387:e=1
\u3000\u3000\u7126\u70b9:(p/2,0)
\u3000\u3000\u51c6\u7ebf\u65b9\u7a0bl:x=-p/2
\u3000\u3000\u9876\u70b9:(0,0)
\u3000\u3000\u901a\u5f84(\u5b9a\u4e49\uff1a\u5706\u9525\u66f2\u7ebf\uff08\u9664\u5706\u5916\uff09\u4e2d\uff0c\u8fc7\u7126\u70b9\u5e76\u5782\u76f4\u4e8e\u8f74\u7684\u5f26)\uff1a2P [\u7f16\u8f91\u672c\u6bb5]4.\u5b83\u7684\u89e3\u6790\u5f0f\u6c42\u6cd5\uff1a \u3000\u3000\u4ee5\u7126\u70b9\u5728X\u8f74\u4e0a\u4e3a\u4f8b
\u3000\u3000\u77e5\u9053P\uff08x0,y0)
\u3000\u3000\u4ee4\u6240\u6c42\u4e3ay^2=2px
\u3000\u3000\u5219\u6709y0^2=2px0
\u3000\u3000\u22342p=y0^2/x0
\u3000\u3000\u2234\u629b\u7269\u7ebf\u4e3ay^2=(y0^2/x0)x [\u7f16\u8f91\u672c\u6bb5]5.\u629b\u7269\u7ebf\u7684\u5149\u5b66\u6027\u8d28: \u3000\u3000\u7ecf\u8fc7\u7126\u70b9\u7684\u5149\u7ebf\u7ecf\u629b\u7269\u7ebf\u53cd\u5c04\u540e\u7684\u5149\u7ebf\u5e73\u884c\u629b\u7269\u7ebf\u7684\u5bf9\u79f0\u8f74\u3002 [\u7f16\u8f91\u672c\u6bb5]6.\u629b\u7269\u7ebf\u7684\u4e00\u6bb5\u7684\u9762\u79ef\u548c\u5f27\u957f\u516c\u5f0f \u3000\u3000 \u9762\u79ef Area=2ab/3
\u3000\u3000\u5f27\u957f Arc length ABC
\u3000\u3000=\u221a(b^2+16a^2 )/2+b^2/8a ln((4a+\u221a(b^2+16a^2 ))/b) [\u7f16\u8f91\u672c\u6bb5]7.\u5176\u4ed6 \u3000\u3000\u629b\u7269\u7ebf\uff1ay = ax^2 + bx + c \uff08a\u22600)
\u3000\u3000\u5c31\u662fy\u7b49\u4e8eax \u7684\u5e73\u65b9\u52a0\u4e0a bx\u518d\u52a0\u4e0a c
\u3000\u3000a > 0\u65f6\u5f00\u53e3\u5411\u4e0a
\u3000\u3000a < 0\u65f6\u5f00\u53e3\u5411\u4e0b
\u3000\u3000c = 0\u65f6\u629b\u7269\u7ebf\u7ecf\u8fc7\u539f\u70b9
\u3000\u3000b = 0\u65f6\u629b\u7269\u7ebf\u5bf9\u79f0\u8f74\u4e3ay\u8f74
\u3000\u3000\u8fd8\u6709\u9876\u70b9\u5f0fy = a\uff08x-h\uff09^2 + k
\u3000\u3000\u5c31\u662fy\u7b49\u4e8ea\u4e58\u4ee5\uff08x-h\uff09\u7684\u5e73\u65b9+k
\u3000\u3000h\u662f\u9876\u70b9\u5750\u6807\u7684x
\u3000\u3000k\u662f\u9876\u70b9\u5750\u6807\u7684y \u6807\u51c6\u5f62\u5f0f\u7684\u629b\u7269\u7ebf\u5728x0\uff0cy0\u70b9\u7684\u5207\u7ebf\u5c31\u662f \uff1ayy0=p(x+x0)
\u3000\u3000\u4e00\u822c\u7528\u4e8e\u6c42\u6700\u5927\u503c\u4e0e\u6700\u5c0f\u503c
\u3000\u3000\u629b\u7269\u7ebf\u6807\u51c6\u65b9\u7a0b:y^2=2px
\u3000\u3000\u5b83\u8868\u793a\u629b\u7269\u7ebf\u7684\u7126\u70b9\u5728x\u7684\u6b63\u534a\u8f74\u4e0a,\u7126\u70b9\u5750\u6807\u4e3a(p/2,0) \u51c6\u7ebf\u65b9\u7a0b\u4e3ax=-p/2
\u3000\u3000\u7531\u4e8e\u629b\u7269\u7ebf\u7684\u7126\u70b9\u53ef\u5728\u4efb\u610f\u534a\u8f74,\u6545\u5171\u6709\u6807\u51c6\u65b9\u7a0by^2=2px y^2=-2px x^2=2py x^2=-2py
\u53cc\u66f2\u7ebf\u6807\u51c6\u65b9\u7a0b\uff1a1.\u7126\u70b9\u5728X\u8f74\u4e0a\u65f6\u4e3a\uff1ax^2/a^2 - y^2/b^2 = 1
2.\u7126\u70b9\u5728Y \u8f74\u4e0a\u65f6\u4e3a\uff1ay^2/a^2 - x^2/b^2 = 1 \u8fd9\u91ccc^2=a^2+b^2
\u7126\u70b9\u5750\u6807\u4e3a(\u00b1c,0)
\u629b\u7269\u7ebf\u6807\u51c6\u65b9\u7a0b:
y2 =2px\uff08p>0\uff09\uff08\u5f00\u53e3\u5411\u53f3\uff09\uff1b
y2 =-2px\uff08p>0\uff09\uff08\u5f00\u53e3\u5411\u5de6\uff09\uff1b
x2 =2py\uff08p>0\uff09\uff08\u5f00\u53e3\u5411\u4e0a\uff09\uff1b
x2 =-2py\uff08p>0\uff09\uff08\u5f00\u53e3\u5411\u4e0b\uff09\uff1b
\u7126\u70b9\u5750\u6807\u4e3a(p/2,0)
\u692d\u5706\uff1a1.\u5f53\u7126\u70b9\u5728x\u8f74\u65f6,\u692d\u5706\u7684\u6807\u51c6\u65b9\u7a0b\u662f\uff1ax^2/a^2+y^2/b^2=1,(a>b>0)\uff1b
2.\u5f53\u7126\u70b9\u5728y\u8f74\u65f6,\u692d\u5706\u7684\u6807\u51c6\u65b9\u7a0b\u662f\uff1ay^2/a^2+x^2/b^2=1,(a>b>0)\uff1b
\u8fd9\u91ccc^2=a^2-b^2 \u7126\u70b9\u5750\u6807\u4e3a(\u00b1c,0)
(x^2/a^2)-(y^2/b^2)=1 （焦点x轴） (y^2/a^2)-(x^2/b^2)=1 （焦点y轴）：双曲线
y^2=2px （焦点x正）y^2=-2px（焦点x负） x^2=2py（焦点y正） x^2=-2py（焦点y负）：抛物线
准线：椭圆和双曲线：x=(a^2)/c
抛物线：x=p/2 （以y^2=2px为例）
焦半径：
椭圆和双曲线：a±ex （e为离心率。x为该点的横坐标，小于0取加号，大于0取减号）
抛物线：p/2+x （以y^2=2px为例）
以上椭圆和双曲线以焦点在x轴上为例。
弦长公式：设弦所在直线的斜率为k,则弦长=根号[（1+k^2）*(x1-x2)^2]=根号[（1+k^2）*((x1+x2)^2-4*x1*x2)] 用直线的方程与圆锥曲线的方程联立，消去y即得到关于x的一元二次方程，x1，x2为方程的两根，用韦达定理即可知x1+x2和x1*x2，再代入公式即可求得弦长。
抛物线通径=2p
抛物线焦点弦长=x1+x2+p 用焦点弦的方程与圆锥曲线的方程联立，消去y即得到关于x的一元二次方程，x1，x2为方程的两根
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