85大學聯考數學（自然組）選填第17題詳解

1. 由於 $H$ 為 $\Delta ABC$ 的垂心，因此 $\overset{\large\rightharpoonup}{BH} \perp \overset{\large\rightharpoonup}{AC}$ 且 $\overset{\large\rightharpoonup}{CH} \perp \overset{\large\rightharpoonup}{AB}$。 由此可得內積關係式： $$\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AH} = \overset{\large\rightharpoonup}{AC} \cdot \overset{\large\rightharpoonup}{AH} = \overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC}.$$ 2. 利用餘弦定理求出 $\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC}$： $$ \begin{aligned} \overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC} &= \overline{AB} \cdot \overline{AC} \cos A \\ &= \dfrac{\overline{AB}^2 + \overline{AC}^2 - \overline{BC}^2}{2} \\ &= \dfrac{8^2 + 4^2 - (2\sqrt{13})^2}{2} \\ &= \dfrac{64 + 16 - 52}{2} = 14. \end{aligned} $$ 因此，有： $$\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AH} = 14 \ \text{且} \ \overset{\large\rightharpoonup}{AC} \cdot \overset{\large\rightharpoonup}{AH} = 14.$$ 3. 將 $\overset{\large\rightharpoonup}{AH} = x\overset{\large\rightharpoonup}{AB} + y\overset{\large\rightharpoonup}{AC}$ 代入上列兩式： （1） $$ \begin{aligned} \overset{\large\rightharpoonup}{AB} \cdot (x\overset{\large\rightharpoonup}{AB} + y\overset{\large\rightharpoonup}{AC}) &= 14 \\ &\implies x |\overset{\large\rightharpoonup}{AB}|^2 + y (\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC}) = 14 \\ &\implies 64x + 14y = 14 \\ &\implies 32x + 7y = 7. \end{aligned} $$ （2） $$ \begin{aligned} \overset{\large\rightharpoonup}{AC} \cdot (x\overset{\large\rightharpoonup}{AB} + y\overset{\large\rightharpoonup}{AC}) &= 14 \\ &\implies x (\overset{\large\rightharpoonup}{AB} \cdot \overset{\large\rightharpoonup}{AC}) + y |\overset{\large\rightharpoonup}{AC}|^2 = 14 \\ &\implies 14x + 16y = 14 \\ &\implies 7x + 8y = 7. \end{aligned} $$ 4. 聯立兩方程式： $$ \left\{ \begin{aligned} 32x + 7y &= 7,\\ 7x + 8y &= 7. \end{aligned} \right. $$ 相減得 $25x - y = 0 \Rightarrow y = 25x$。 代入得 $7x + 8(25x) = 7 \Rightarrow 207x = 7 \Rightarrow x = \dfrac{7}{207}$。 則 $y = 25 \times \dfrac{7}{207} = \dfrac{175}{207}$。 故數對 $(x,y) = (\dfrac{7}{207}, \dfrac{175}{207})$。