直線的參數方程式-例20081010-2356

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$A=\left( 5,-2 \right),B=\left( 1,4 \right),P=\left( x,y \right)\in \overline{AB}$
$\left\{ \begin{align} & x=5+\Delta x\cdot t=5+\left( 1-5 \right)t=5-4t \\ & y=-2+\Delta y\cdot t=-2+\left( 4+2 \right)t=-2+6t \\ & 0\le t\le 1 \\ \end{align} \right.$
$\therefore 3x+y-2=3\times \left( 5-4t \right)+\left( -2+6t \right)-2$
$=15-12t-2+6t-2=11-6t\,$
$0\le t\le 1\Rightarrow -6\le -6t\le 0\Rightarrow 5\le 11-6t\le 11$
$\therefore 5\le 3x+y-2\le 11\#$
$2x^{2}-y^{2}+1=2\times \left( 5-4t \right)^{2}-\left( -2+6t \right)^{2}+1$
$=2\times \left( 25-40t+16t^{2} \right)-\left( 4-24t+36t^{2} \right)+1$
$=50-80t+32t^{2}-4+24t-36t^{2}+1\,$
$=-4t^{2}-56t+47=-4\times \left( t^{2}+14t \right)+47$
$=-4\times \left( t^{2}+14t+7^{2} \right)+4\times 7^{2}+47$
$=-4\times \left( t+7 \right)^{2}+243$
$0\le t\le 1\Rightarrow -4\times \left( 1+7 \right)^{2}+243\le -4\times \left( t+7 \right)^{2}+243\le -4\times \left( 0+7 \right)^{2}+243$
$\Rightarrow -13\le -4\times \left( t+7 \right)^{2}+243\le 47$
$\therefore -13\le 2x^{2}-y^{2}+1\le 47\#$