﻿ 拋物線，焦弦的兩端點坐標之性質(頁 1) - 高中的數學 - IV：線性代數 - Math Pro 數學補給站

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### 拋物線，焦弦的兩端點坐標之性質

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1. $tk=-1.$
2. $x_1x_2 = c^2 且 y_1y_2 = -4c^2.$
3. $\overline{AB}=c\left(t-k\right)^2=c\left(t+\frac{1}{t}\right)^2.$
4. $\triangle VAB 面積 = c^2\left|t-k\right|=c^2\left|t+\frac{1}{t}\right|.$

1.

$\overrightarrow{FA}=(ct^2 - c, 2ct),\overrightarrow{AB}=(ck^2 - ct^2, 2ck-2ct)$

$(ct^2 - c) : 2ct = (ck^2 - ct^2) : (2ck-2ct)$

$tk=-1.$

2.
$x_1x_2 = ct^2\cdot ck^2 = c^2 (tk)^2 = c^2$

$y_1y_2 = (2ct)\cdot(2ck) = 4c^2(tk) = -4c^2.$

3.

$\overline{AB} = \overline{AF} + \overline{FB}= A到準線的距離 + B 到準線的距離$
$=(x_1 + c) + (x_2+c)=(ct^2 + c) + (ck^2 +c)$
$= c\left(t^2 + k^2 +2\right) = c\left(t^2 + k^2 -2tk\right)$
$=c(t-k)^2 = c(t + \frac{1}{t})^2.$

4. $\triangle VAB 面積 = \frac{1}{2} | \left| {\begin{array}{*{20}c} {\overrightarrow {VA} } \\ {\overrightarrow {VB} } \\ \end{array}} \right| | = \frac{1}{2} | \left| {\begin{array}{*{20}c} {ct^2 } & {2ct} \\ {ck^2 } & {2ck} \\ \end{array}} \right| |$
$=c^2\left|tk(t-k)\right| = c^2\left|\left(-1\right)\left(t-k\right)\right|$
$= c^2\left|t-k\right| = c^2 \left|t + \frac{1}{t}\right|.$