已知空间两异面直线间的距离为[tex=1.357x1.214]lSQmNYxbPA9v56YNpnqgNA==[/tex] 夹角为[tex=1.286x1.214]Xotn2yxe0uPWz/8EYVYFJA==[/tex]过这两直线分别作 平面，并使这两平面互相垂直，求这样的两平面交线的轨迹.
[b]解法一[/b] 取两异面直线[tex=0.714x1.214]/9VPWMAe45DE7o41XjpmwQ==[/tex]与 [tex=0.714x1.214]6u7HMOUU+3yIS4FoZkCfDg==[/tex] 的公垂线为[tex=0.5x0.786]C7x+w8+jOPZzxFrGGne6Dw==[/tex]轴，公垂线的中点为坐标原点， [tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex]轴与[tex=1.643x1.214]+FfH2Eh4kHr2t34PTpkq/Q==[/tex]成等角建立直角坐标系. 那么两异面直线[tex=0.714x1.214]/9VPWMAe45DE7o41XjpmwQ==[/tex] 与[tex=0.714x1.214]6u7HMOUU+3yIS4FoZkCfDg==[/tex]的方程为[tex=23.071x2.357]7CaM20IuNtyWrSn+waM/PNFONAhibnACSbCd0uhHh/WQV0TDwVIcVV5efCKcHGuCA88x6MK34OSlS001tTYpXujOzRGgvmME3Ii1q4vpn6j7HkBxeivcFw5BlmlY8WyoASR5mEYOa5f62s+GdHZvPCAZYkdMTY+XD1YlDi/t4EDT0ssDdrXlZ5u/3GftkVjZ[/tex]或化[tex=0.714x1.214]/9VPWMAe45DE7o41XjpmwQ==[/tex]与[tex=0.714x1.214]6u7HMOUU+3yIS4FoZkCfDg==[/tex]的方程为一般式[tex=23.0x2.786]VNVlbbp9pWdAvXgfYu2ErGo/jl2YdySEMByUnwu5x85LOszrh3njpoUJBlShot9WeDKM3qSmNAfjPMWZI/R5PF9AZzo0D8Q+jAgNSmqbUJ96qhGWSEVUtODEFYK28HOtc2Ab9sAlCbCrpCBywawqFg2391S8F71FsxsBJR1WiBOH+qgqd7N0Ip6ZaI4rATBwP5U98JwGUsrdvz9qU5PWN5KwW4OdnsulcYXVi5lk2PPLNoOTdre2T3MQs4+dZ5g4[/tex]利用平面束，通过两异面直线 [tex=0.714x1.214]/9VPWMAe45DE7o41XjpmwQ==[/tex] 与[tex=0.714x1.214]6u7HMOUU+3yIS4FoZkCfDg==[/tex]的平面分别为[tex=35.714x1.643]rZM5/OPAdr7aX+kNl9iwpCzwnlcfUXEKJuq//bDoX4fPfQAkETb0HJlgL4ajXdIcFa/nicWU1NvPep6if5lvxuN2AFZEywA+ToQ5XEU/dgVj+jV0qDs0+A/QpxVOwHJmFdtLjXsMtvC1ECArU+uL1L7f/QDHZT1ND4MjOo1epAsAkCPP1ZQkRpwJC464rpCcOy5ZS0KjwOd36mimN4LUnzu1iK027zB9nCTJmp0XK1M=[/tex]因为[tex=6.357x1.214]AIdwTY5iHr/ZS6yQmYizmXOXPm8WLtYPaHg9UnM5PB6sNINWwMYoFtqNMViKrVG5[/tex], 所以得[tex=19.857x1.429]zpMvPR07/Z2Wqp8npOzoqmtFy9EjITDO4+woHdV+TPkV5aijSao65GygRrBWLCJC/aBG5mHh3WecoFfqYy3gHVo7Xh9xjpOR61R2rISjszT+DOS8ATp52VGr9NdaEIU6T2kw4M8IbLn85QFLjDikKbmnRlFoz6DaxQidqAtZ4QYzJoxZXNozN9wYI+P7q650[/tex]亦即[tex=8.0x2.643]atBEcP9blgehQNrv496FKPwHK6/63cyoRUpc3UImxn09Njo2oO3JY6eFGFcE5SRb/pbIxp8M+My3t5/wYLambQ==[/tex]由式[tex=2.571x1.357]pLCcHl3JR0PJLB9kH2hoRg==[/tex]，式[tex=2.571x1.357]cRx1AluNz8eLILSRbjgeaQ==[/tex] 得[tex=18.071x3.071]rZM5/OPAdr7aX+kNl9iwpCSACM31meLTuNf6kSlpNyEpXHlIoo89KbShNWiWn4jpIzZnqBAw5LdfC68javAoUYUffZHdGb573Ow5FNrzTzSXy16nwbYZYsO++Uoy5k7Tv4HMXrShWj0ImwtIaypkL449bNePgjsVj4zpNvMILBJf4TjRSZKNeWthJkjqaW1x9Z7IH3ClO8T9Y8miy6qiAiw/MzzgjCDvvxaEa87XXfc=[/tex]代入式 [tex=2.571x1.357]mGTT02kVJ9DfG9akrW3PKQ==[/tex] 得[tex=15.714x2.714]TlQ3xZhLZ5l74++YNKU+lmEvidf3vOi4ErvYK/j5mSPj/ru1HpoMCWTxdoqJlBUVKOFOLI33WcvhpoNtU9w4F7SjNJdHNLcovGrrnPR1ZDlBxOqFqTfPMdNtqgrIZPYs7DAKevlW6n0UhQUZCQfqcw==[/tex]即[tex=17.071x1.429]5MHMX2bolLX4kf1c93JbLq6yC+2Yz8GcvPEIhQszXy9qyaOiOLqHExr4bVc373Mq7UBhVjduiAETiPxEvcMiwtW376qLwB7uJTGRUWlVbucRFg8Ptb9sLG8LMo7kjeB3[/tex]当[tex=3.071x1.786]7FozOcU7KGcSffoteg2dkiCCPd7EIaSHzNDNtxCyIvFgSuWo1OvRzydSamDpPg9s[/tex]时，轨迹为单叶双曲面[tex=11.786x3.0]45s2l+iyQPtlZRGZKSYyJF24zMvKPsx4OflGWusBAAfWAozy8W5KI4aBXNlllN94rs6a8U1MNOUf+St/mfMSZ7uyvttJpmStyM4QKiHj3DlkewB0aU7Vmhga6flGcK4gGLUetsgiiDqfrMkiMyrRs77RvtjDKHz5GWzQ7GIS8xFc+0DX+ol9+Pj1pH8DX8zycXb4EAc0di86Zi1vW7p2iw==[/tex]当[tex=2.786x2.143]H+cScJHrJ00e/hjq5VBp3ALQqh1qu1m8xGx5ifSD8Rg=[/tex] 时[tex=12.571x2.643]iqWAFXziFPlEk198uFS8UX7PrF5prWTDLy0AplOwTGjzearO7QRrl2oAhc5xrs3FUU6bQKYm64eeZG1L84hVeGsaEvkIDcHA4Q65y+tYSF4=[/tex]轨迹为两相交平面[tex=4.429x1.429]Ag5Tq51uyffLQ5+w4p377cDwlLNL1rwyw4DHglOrXnY=[/tex]即[tex=3.571x1.214]jADnb5MNwXjSW+jd7PJhKg==[/tex][b]解法二[/b] 因为[tex=1.643x1.214]+FfH2Eh4kHr2t34PTpkq/Q==[/tex]方向向量分别为[tex=18.143x1.357]A99Nj05OGPQPk/SlYqOGCP2e+0KAUjqI9SOBES2Nr7GlGP9PAUwdbvcpKIQCfox86Il63Lwyy//IEEAKD8KaSddQo+Irp1KN/+Z8Wmca/Wt1pyp7afFSW9O3Tl8Pzl1v[/tex]过[tex=0.714x1.214]/9VPWMAe45DE7o41XjpmwQ==[/tex] 的平面为[tex=10.929x1.357]4G/4el6d3qTt5An3hQILDqG0ZOAU653qGx4B7DEOVm4=[/tex]其中[tex=7.786x1.143]zO4BIcvCTE5NVHe5soUgMuvMt35Kx78V48NSt4sqtR0=[/tex], 从而得[tex=29.429x4.5]qeiYnKXLEhyhuGRg8yLtr5byHiRsqjJK+EtMWumZIacLgn4YFJ6Zb9U9IzbBwdxSzY+cKgGyh0Vg+JVoCqBoF3Lh2aVVvWVLBLKxZQ7D/1Xm2iueoxHugOgssFkmXSAF/4fZGtQra3KmukpQkiJ0cAjnrrSHFIeabgg9NavdZPqECgxWWrcrPHB1EKxpwuEdw3LytH+9R5jNtQx2W35qcY0mSs+bQyoj7m5vE6wVH8LLLDnT6+epwafEZa2zWjLI8R2cGa0Z6zBG7KBmxoZLWLNfkUGSgJz6nfU/yhPRmLHJlKpwPfmZMoqS601J2k3rSf5sNY5vLoyBpXMMuK2vCErbOy0yqrN37WdzCEgPJHh0iqiFUinBavWtH1ZQrqs8FJqe0saPfNNG6DAoRJdNH2MVy7tDB5vsHuh99VE+vuJf3N/dRNhLQdYnBh/uzPJKwy8+mJmkjC989TnBSUgegA==[/tex]过 [tex=0.714x1.214]6u7HMOUU+3yIS4FoZkCfDg==[/tex]的平面为[tex=11.286x1.429]VQhL5cW17zTNVbik127m/tBpEREDpu31Kju/r5A6tzeQ6/wQ78k5Hzs5g+suokjQif/1NHzESWDSjQTTH6xDIg==[/tex]其中[tex=8.571x1.357]n/yVvAsj/HFy4L32FpzyVDbw0visyV+KsdBovFSnXhK8o/g8QjIe5/QrolF9oUFVe23wIz356tWkOK9DKFfKbQ==[/tex]从而得[tex=31.214x4.5]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[/tex]因为[tex=3.214x1.214]AIdwTY5iHr/ZS6yQmYizmdVB81JUTtPuXDXfeOaC/jg=[/tex], 所以[tex=10.643x1.429]RY2MNA5rMYZ1POkJy/HxSnlMTM72SJZmpbsL9lQPWOXvm9HzkK1ot228+kqRg1LeQxYt8QWTWEzBP2xNpvYnPQ==[/tex]将式[tex=2.571x1.357]j2+C20gUSA01DCG0NcK1+g==[/tex]，式[tex=2.571x1.357]JspsS4HCttqiVYgma3vCyQ==[/tex]代入式[tex=2.571x1.357]ZwXvtx3AS5HYROHHYImEhQ==[/tex]即得轨迹方程为[tex=26.357x1.571]nnwnLwBQ+sj95jVFw9GflydOTrqXQXMtMWg4nH8SOuTGDbFogj3E5PDYuEfbvrsflGpXVIYyGun0Y4oe/NdPPTnV9FHsayma3QBhsELYWMLFMBS/W2ozLTtLwbGlbPuE8W2jHU2sIr8p2xYGPx2O6Wqg0I56YYwzCInrvWrviZJwU228efd+08/GFIt9S0nQGi0v5P0NNYdMcgowrZXjgA==[/tex]化简得[tex=17.071x1.429]5MHMX2bolLX4kf1c93JbLq6yC+2Yz8GcvPEIhQszXy9qyaOiOLqHExr4bVc373Mq7UBhVjduiAETiPxEvcMiwtW376qLwB7uJTGRUWlVbucRFg8Ptb9sLG8LMo7kjeB3[/tex]当 [tex=3.071x1.786]7FozOcU7KGcSffoteg2dkiCCPd7EIaSHzNDNtxCyIvFgSuWo1OvRzydSamDpPg9s[/tex]时，即[tex=0.714x1.214]/9VPWMAe45DE7o41XjpmwQ==[/tex]不垂直[tex=0.714x1.214]6u7HMOUU+3yIS4FoZkCfDg==[/tex]时，轨迹为单叶双曲面[tex=11.786x3.0]45s2l+iyQPtlZRGZKSYyJF24zMvKPsx4OflGWusBAAfWAozy8W5KI4aBXNlllN94rs6a8U1MNOUf+St/mfMSZ7uyvttJpmStyM4QKiHj3DlkewB0aU7Vmhga6flGcK4gGLUetsgiiDqfrMkiMyrRs77RvtjDKHz5GWzQ7GIS8xFc+0DX+ol9+Pj1pH8DX8zycXb4EAc0di86Zi1vW7p2iw==[/tex]当 [tex=2.786x2.143]H+cScJHrJ00e/hjq5VBp3ALQqh1qu1m8xGx5ifSD8Rg=[/tex]时，即[tex=2.643x1.214]X9fNkbNXlDvy8bPbboNWKk6vKjN3ro1dluXVhh5jhWk=[/tex]时，[tex=9.357x1.214]yvSuhzVbkl/iT1SyxdVMA/65ryarL+XJSdpy8pStEi1izgTLDK5T6mcizYMRywWu[/tex] 故轨迹为一对相交平面[tex=4.429x1.429]Ag5Tq51uyffLQ5+w4p377cDwlLNL1rwyw4DHglOrXnY=[/tex]即[tex=3.571x1.214]Gj9sjWnj3eouLoIUAUyTww==[/tex]
举一反三
- 已知空间两异面直线间的距离为[tex=1.071x1.0]g6m/nu3UX5mtPlafz5e7rg==[/tex],夹角为[tex=1.143x1.0]Tfa34hm32dA6wcD+i8MGrw==[/tex],过这两条直线分别作平面,并使这两平面相互垂直,求这样两平面交线的轨迹.
- 适当选取坐标系，求轨迹方程：求与两给定直线等距离的点的轨迹，已知两直线之间的距离为[tex=0.571x1.286]mRKL/orzOudCEARA8qn3Kw==[/tex]，夹角为[tex=0.643x1.286]vYiGJJ9TAtvnQmM1PsOB8g==[/tex] .
- 适当选取坐标系，求下列轨迹方程：与两给定的异面直线等距 离的动点轨迹，已知这两条异面直线的距离为 [tex=1.0x1.0]RxXNSJgLZ3lVJhUDgE6BVg==[/tex], 夹角为 [tex=1.5x1.357]55NHKATJVmY//E+aOmMYzg==[/tex] 取公垂线为 [tex=0.5x0.786]C7x+w8+jOPZzxFrGGne6Dw==[/tex] 轴，公垂线段的中点为原点， [tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex] 轴与两直线 成等角).
- 过两交又直线 [tex=3.571x1.214]Cku2M4ntpnLGOkIkk9sCeA==[/tex]作互相平行的两平面 [tex=2.286x1.0]7G+DRyq9DQdwAo7mOI27Xg==[/tex]和 [tex=2.214x1.0]lkW5TGXMmqivG7e49ZDc3w==[/tex] 。[img=790x938]17ae5b30aa399fe.png[/img]
- 已知空间两异面直线[tex=0.714x1.214]/9VPWMAe45DE7o41XjpmwQ==[/tex]和 [tex=1.786x1.214]fFDkEfbsGm8FgIFigbiK/g==[/tex]与 [tex=0.714x1.214]6u7HMOUU+3yIS4FoZkCfDg==[/tex] 不垂直，它们的距 离为[tex=1.214x1.214]i0JiNs94bCXlyWFPBy02mA==[/tex] 夹角为[tex=1.357x1.214]EvQCQFUkpkiFT1d2XoWa7w==[/tex] 分别过 [tex=0.714x1.214]/9VPWMAe45DE7o41XjpmwQ==[/tex] 及 [tex=0.714x1.214]6u7HMOUU+3yIS4FoZkCfDg==[/tex] 作两个互相垂直的平面. 证明交线的轨迹是一个单叶双曲面. (取公垂线为 [tex=0.5x0.786]C7x+w8+jOPZzxFrGGne6Dw==[/tex] 轴，其中点为原点，[tex=0.571x0.786]ZKO2xs0EgSemzoH7MSmYTA==[/tex] 轴与两条直线成等角）
内容
- 0
适当选取坐标系，求下列轨迹的方程：求与两给定直线等距离的点的轨迹，已知两直线之间的距离为[tex=0.571x0.786]c59+3vo0/Vn/FvNRhDRu5g==[/tex]，夹角为[tex=0.643x0.786]SPoVA3bJlgfP9Ek9O4AbuA==[/tex]。
- 1
求通过原点，且垂直于两直线[tex=6.643x1.214]OfW3lwflYqKoFy46Zenx5g==[/tex]与[tex=7.143x1.214]XZN+un3YlTzODRDzw803XQ==[/tex]的平面方程
- 2
求过点 [tex=3.214x1.357]IJZ7WbMh0EydO3A6WuE6sg==[/tex] 且与两平面 [tex=4.643x1.143]XoQ2luuo+BtyzXhZI2+nHA==[/tex] 和 [tex=4.071x1.214]lOkbb6Rz1dJi1t//6qLTvQ==[/tex] 都平行的直线方程.
- 3
过直线[tex=1.571x1.0]JLMbVw4e37VvhkU494+8Ew==[/tex]作平面垂直已知平面。[img=472x426]17ab962346ceac1.png[/img]
- 4
分别在下列条件下确定[tex=2.714x1.214]j4pPwaSjeHdr5apw5WTadQ==[/tex]的值：使[tex=15.857x1.286]CopJvmI8/V+O7lzs0CHNmXHeUmefKG2xAVfLQhJ2RtZaXSMufPH/65cTkSQVDkn9[/tex]表示两互相垂直的平面.