将下列二次圆锥曲线方程化为标准形，并指出它们的形状：[tex=17.571x1.5]gMtHmiRS+HWEnGPFl4SgGV6gGUIgIhMbwcCVAb6yr1IImtTd06qdKQ3YJdfO/Qry[/tex]

2022-07-02

将下列二次圆锥曲线方程化为标准形，并指出它们的形状：[tex=17.571x1.5]gMtHmiRS+HWEnGPFl4SgGV6gGUIgIhMbwcCVAb6yr1IImtTd06qdKQ3YJdfO/Qry[/tex]

答案：

先将二次型[tex=6.5x1.429]p1SCVtb3wPtqeutJJWnlKrvRWFLEA4jNKl/259u0VtY=[/tex]用正交变换化为标准形二次型矩阵[tex=7.857x2.786]YHIKtHtTy6YeetIBukGWJ/+4CPibC+H1hzFh9VOwgL23z42PQ1nUeB6xXFF9LbiNl+TXjt5aPVefbp+3LbL5KDfQ8LzkjyjhceH2Zm5nVkA=[/tex]，解特征方程[tex=7.857x2.786]Uyz5s0rmQIddjb5Jc2T/YekF+dXnWk4JooFE8jCyE4YswP0iffFrYBOxuTU4fWxH7JF3V0IUQOE2yG8OmnVm2p/K+NcSh8ocud8a3rCyaaU=[/tex]，得其特征值为[tex=4.929x1.214]Zsi8Crqj3p4Ruo9U4Ix8DQlSkMbGMslsLRA06xRfQdI=[/tex]，相应地，特征向量为[tex=11.5x2.786]qulE2au0sCsC2RUF6/a3J3NtG8/hCqny8tlhGzhMKwNXjiC2dQYvK4q24/2b3cLF3fRahJXLNpyS7tY2m8uoKn0qABT1D2wA/ja9HWcjfvtiKV5Uy5Q+l7TLr4psgIeV1yz1HaAXpLmItJa0rcDKFrBcXGyF8cIqNMyUwXJaR4KgK7IWw2UgMWDL4wDYJ1xnEZlg7Lnot4qexdjOh8oy8A==[/tex]，正交单位化得[tex=14.786x2.786]4IwCB7mUZT1bCJOXMr2ql/WD4Jr7xQa6wNu3u7bSc76xJ9rEuLxKXl7IFZ5tH/G9MwIeBNxAuHGjHWvlCGi405QH2PeyyFv8aZGyoVkr3vEYtR0HXlFh1X+K3LWDB0uc/YZ4liv7TpqI2nRO0On/huYGXRNtN1h9YnG+kf9SzgyAc9aiSjqNLgP/eFuZFIO7LOMgOK002CC4LD96oTnpxsVTrZmz7XwUd2MOftkYcvw=[/tex].作正交变换[tex=14.214x2.786]075gCzZzsMRb6HYXYk9X91xtdU1wg4Ye92uOi7+SMR/bto4PrzAtGjJBH8AvvMII7SqJytvW20sIBrN0eNMVSwmoj1ytQiZqySQY9c+rpW+9kZ98lNtypp+TVKJmkUSG/hYuBl5dkRbHlQKBbJsusK00EDrbjXw5qDvVdgw1C61ZsXPTJJOhj5JyANalY21BFkGoEiy+wyiqiSbJeQ33++qR6Jh0gJNHemKLDeQmSMlnFYS2tY9NMx/emUvu92Xw[/tex]，将原二次圆锥曲线为[tex=14.357x2.643]MjzZASYdRS6vpj7MIC4Wg4xp4I25iXZe7LeqNmoGO/i4KS9iQhVA2WrwS4pU2F7rqoQTAu9ix3eXWoXa66ZiEcuHVI5ic34qOejH1G3NJ/g=[/tex]，配方后，得标准形[tex=16.571x2.929]3C1D2s+vpDEROnZg6lK/glMPyQ3+RuQS8ERhNOA5/b2Sn76xJJeFvK9qEnd0i9qA/GsA8/Bygc51478ip77mSs3oOGB6GBIyNMuzCtZ8DsjqxSgmSsYh6dTjdK5/p7106KS7VY7c+fvELSQuQ4DIIw==[/tex]，所以它表示的是一个椭圆。

举一反三

内容

0
将下列直线方程化为标准方程及参数方程：[tex=9.143x2.786]GE56u9QCDTqcLxZ66HADyp3zrrfvmt9jpVQbB6Oz0yNMhJHKmzuxfrGOOPOMdQipzc9WPHcIvnBfyBt0j3wYLFVYuYuBs8wdVbXX9fq7jc0=[/tex]
1
将下列直线方程化为标准方程及参数方程：[tex=7.071x2.786]GE56u9QCDTqcLxZ66HADyla8DilgXcC0583fIdmW9remmdi9DkL+iXZctpfNXncayglkZtaecIsfaNQOZChczhjewKomM90ccgFPVI08YWw=[/tex].
2
求一个正交变换[tex=8.143x3.357]jcCMHflCR8OS9TosV6N5vBJLPA1uBQTxcynNa8WyQjcsEDIpJ03M0BX+8NK/C3EVtj3hFZwtZLEqfpWEBPgb9bqS7Ui18ZKUcUb7W7LFhJFfQVW2Hnc9/Y+y63FclgXj8WP2VHFOREMkIinBpyi66g==[/tex],将二次曲面方程[tex=13.0x1.429]7PPpgWZDCTQdo4vyO8xzv6JyN1U5MOyW2LUmbFrh/zjJwEDOKkZtlXI/lN+8JJGa[/tex]化为标准形方程,并问该二次曲面是什么类型的曲面.
3
将下列方程化为标准形式：[tex=5.286x1.5]HBzgdCGw8A8Xdeq1P4xsjZGuIwB/VuoQfftmyIeOnNA=[/tex].
4
已知实二次型[tex=8.429x1.5]CKlOGn/4oc+CIjd/NrEXYL4dkmrBlW2GCC51Nn4jxKDKCw4XacjR6Wkq7aCfgBox[/tex]经正交变换[tex=2.857x1.0]NGl8c6nTOCjuPJKs2cOxRw==[/tex]可化为标准形[tex=5.571x1.5]BxX+JKGKByN8807bHgu+T1pktokv0l3dVZx8FO3ro+VWSdtzbyfx30MD7Apuy3Zt[/tex],则矩阵[tex=4.0x1.357]EBNDcMm1xMDWg08FZWWY1Q==[/tex][input=type:blank,size:6][/input].