我的习惯是用柯西及其类似,写起来比较对称:
\begin{align*}
(a^2 + b^2) (c^2 + d^2) = (a c + b d)^2 + (a d - b c)^2 &\riff (a^2 + b^2) (c^2 + d^2) \geqslant (a c + b d)^2 ,\\
(a^2 - b^2) (c^2 - d^2) = (a c - b d)^2 - (a d - b c)^2 &\riff (a^2 - b^2) (c^2 - d^2) \leqslant (a c - b d)^2 ,\\
\end{align*}
所以有
\begin{align*}
\left(\frac{x_0^2}{a^2}+\frac{y_0^2}{b^2}\right)
\left(\frac{x_T^2}{a^2}+\frac{y_T^2}{b^2}\right)
\geqslant\left(\frac{x_0x_T}{a^2}+\frac{y_0y_T}{b^2}\right)^2 & \riff \frac{x_T^2}{a^2}+\frac{y_T^2}{b^2}\geqslant1,\\
\left(\frac{x_0^2}{a^2}-\frac{y_0^2}{b^2}\right)
\left(\frac{x_T^2}{a^2}-\frac{y_T^2}{b^2}\right)
\leqslant\left(\frac{x_0x_T}{a^2}-\frac{y_0y_T}{b^2}\right)^2 &\riff \frac{x_T^2}{a^2}-\frac{y_T^2}{b^2}\leqslant1.
\end{align*}作者: zhcosin 时间: 2017-10-12 07:36