951

已知 ${\displaystyle A(x_1,y_1),B(x_2,y_2)}$ 为抛物线 ${\displaystyle y^2=8x}$ 上两个不同的动点，且满足 ${\displaystyle y_1{y}_2=-16}$, 则 ${\displaystyle |x_1+y_1+2|+|x_2+y_2+2|}$ 的最小值为 ${\displaystyle \mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}$.

对 ${\displaystyle i=1,2}$, ${\displaystyle y_i^2=8x_i}$, 则$$\begin{array}{r}{x}_i+y_i+2=\frac{y_i^2}{8}+y_i+2=\frac{1}{8}(y_i+4{)}^2\ge 0,\end{array}$$
从而摘去绝对值后$$\begin{array}{r}\text{原式}=\frac{1}{8}(y_1^2+y_2^2)+(y_1+y_2)+4=\frac{1}{8}(y_1+y_2{)}^2+(y_1+y_2)+8\ge 6,\end{array}$$
取等时${\displaystyle y_1+y_2=-4}$.