下册 7.5 多元函数微分的应用 第36题

\section*{解题过程：} （1）曲线在点 $P\left(x_{0}, y_{0}\right)$ 处的切线方程及法线方程分别为 $$ F_{x}\left(x_{0}, y_{0}\right)\left(x-x_{0}\right)+F_{y}\left(x_{0}, y_{0}\right)\left(y-y_{0}\right)=0, F_{y}\left(x_{0}, y_{0}\right)\left(x-x_{0}\right)-F_{x}\left(x_{0}, y_{0}\right)\left(y-y_{0}\right)=0 . $$ 记 $L(x, y, \lambda)=x^{2}+y^{2}+\lambda F(x, y)$ ．曲线上到 $O(0,0)$ 最近的点 $P\left(x_{0}, y_{0}\right)$ 满足如下的条件： $$ L_{x}(x, y, \lambda)=2 x+\lambda F_{x}(x, y)=0, L_{y}(x, y, \lambda)=2 y+\lambda F_{y}(x, y)=0, L_{\lambda}(x, y, \lambda)=F(x, y)=0 . $$ 由此可知 $\displaystyle \frac{x_{0}}{F_{x}\left(P_{0}\right)}=\frac{y_{0}}{F_{y}\left(P_{0}\right)}$ ．这表明原点满足过点 $P_{0}\left(x_{0}, y_{0}, z_{0}\right)$ 的法线方程，即 $O P$ 为曲线过点 $P\left(x_{0}, y_{0}\right)$ 的法线，从而 $O P$ 与曲线过点 $P\left(x_{0}, y_{0}\right)$ 的切线正交． （2）记 $L(x, y, \lambda)=\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\lambda F(x, y)$ ．曲线上到 $\left(x_{0}, y_{0}\right)$ 最近的点 $\left(x_{1}, y_{1}\right)$ 满足如下 的条件： $$ L_{x}(x, y, \lambda)=2\left(x-x_{0}\right)+\lambda F_{x}(x, y)=0, L_{y}(x, y, \lambda)=2\left(y-y_{0}\right)+\lambda F_{y}(x, y)=0, L_{\lambda}(x, y, \lambda)=F(x, y)=0 $$ 由此知 $$ F_{x}\left(x_{1}, y_{1}\right): F_{y}\left(x_{1}, y_{1}\right)=\left(x_{1}-x_{0}\right):\left(y_{1}-y_{0}\right) $$ 即过点 $P\left(x_{0}, y_{0}\right)$ 及点 $\left(x_{1}, y_{1}\right)$ 的直线为法线，所以曲线 $\Gamma$ 过点 $\left(x_{1}, y_{1}\right)$ 的法线方程为 $$ \frac{x-x_{0}}{x_{1}-x_{0}}=\frac{y-y_{0}}{y_{1}-y_{0}} $$ （3）曲面在 $P_{0}\left(x_{0}, y_{0}, z_{0}\right)$ 处的法线方程为 $\displaystyle \frac{x-x_{0}}{F_{x}\left(P_{0}\right)}=\frac{y-y_{0}}{F_{y}\left(P_{0}\right)}=\frac{z-z_{0}}{F_{z}\left(P_{0}\right)}$ ． 记 $L(x, y, z, \lambda)=x^{2}+y^{2}+z^{2}+\lambda F(x, y, z)$ ．曲面 $F(x, y, z)=0$ 上离原点最近的点 $P_{0}\left(x_{0}, y_{0}, z_{0}\right)$满足如下的条件： $$ \left\{\begin{array}{l} L_{x}(x, y, z, \lambda)=2 x+\lambda F_{x}(x, y, z)=0 \\ L_{y}(x, y, z, \lambda)=2 y+\lambda F_{y}(x, y, z)=0 \\ L_{z}(x, y, z, \lambda)=2 z+\lambda F_{z}(x, y, z)=0 \\ L_{\lambda}(x, y, z, \lambda)=F(x, y, z)=0 \end{array}\right. $$ 由此知 $\displaystyle \frac{x_{0}}{F_{x}\left(P_{0}\right)}=\frac{y_{0}}{F_{y}\left(P_{0}\right)}=\frac{z_{0}}{F_{z}\left(P_{0}\right)}$ ．这表明原点满足过点 $P_{0}\left(x_{0}, y_{0}, z_{0}\right)$ 的法线方程。 （4）记 $L(x, y, z, \lambda)=\left(x-b_{1}\right)^{2}+\left(y-b_{2}\right)^{2}+\left(z-b_{3}\right)^{2}+\lambda F(x, y, z)$ ．曲线上到 $B\left(b_{1}, b_{2}, b_{3}\right)$ 最近的点 $A\left(a_{1}, a_{2}, a_{3}\right)$ 满足如下的条件： $$ \left\{\begin{array}{l} L_{x}(x, y, z, \lambda)=2\left(x-b_{1}\right)+\lambda F_{x}(x, y, z)=0 \\ L_{y}(x, y, z, \lambda)=2\left(y-b_{2}\right)+\lambda F_{y}(x, y, z)=0 \\ L_{z}(x, y, z, \lambda)=2\left(z-b_{3}\right)+\lambda F_{z}(x, y, z)=0 \\ L_{\lambda}(x, y, z, \lambda)=F(x, y, z)=0 \end{array}\right. $$ 由此知 $F_{x}(A): F_{y}(A): F_{z}(A)=\left(a_{1}-b_{1}\right):\left(a_{2}-b_{2}\right):\left(a_{3}-b_{3}\right)$ 。这表明过点 $B\left(b_{1}, b_{2}, b_{3}\right)$ 及点 $A\left(a_{1}, a_{2}, a_{3}\right)$的直线为法线，所以曲面 $\Sigma$ 在点 $A$ 的法线方程为 $$ \frac{x-a_{1}}{b_{1}-a_{1}}=\frac{y-a_{2}}{b_{2}-a_{2}}=\frac{z-a_{3}}{b_{3}-a_{3}} $$ 在点 $A\left(a_{1}, a_{2}, a_{3}\right)$ 处的切平面方程为 $$ \left(b_{1}-a_{1}\right)\left(x-a_{1}\right)+\left(b_{2}-a_{2}\right)\left(y-a_{2}\right)+\left(b_{3}-a_{3}\right)\left(z-a_{3}\right)=0 $$ （5）设 $G(x, y, z)=F(x, y, z)-1$ ，则 $G_{x}=F_{x}(x, y, z), G_{y}=F_{y}(x, y, z), G_{z}=F_{z}(x, y, z)$ ． 在 $P_{0}\left(x_{0}, y_{0}, z_{0}\right)$ 的切平面方程为 化简得 $$ \begin{gathered} F_{x}\left(P_{0}\right)\left(x-x_{0}\right)+F_{y}\left(P_{0}\right)\left(y-y_{0}\right)+F_{z}\left(P_{0}\right)\left(z-z_{0}\right)=0 \\ F_{x}\left(P_{0}\right) x+F_{y}\left(P_{0}\right) y+F_{z}\left(P_{0}\right) z=F_{x}\left(P_{0}\right) x_{0}+F_{y}\left(P_{0}\right) y_{0}+F_{z}\left(P_{0}\right) z_{0} \end{gathered} $$ 把 $x F_{x}+y F_{y}+z F_{z}=n$ 代人上式得 $x F_{x}\left(P_{0}\right)+y F_{y}\left(P_{0}\right)+z F_{z}\left(P_{0}\right)=n$ ．