回覆 10# yuen1008 的帖子
3.
\(\displaystyle a>\frac{1}{2}\)，拋物線\(y=ax^2+2\)之頂點為\(V\)且與\(y=-x+4a\)相交於\(B\)、\(C\)兩點。已知\(\Delta VBC\)面積為\(\displaystyle \frac{72}{5}\)，試求\(a\)之值。
[解答]
令\(x_1,x_2\)為\(ax^2+2=-x+4a\)的兩根，\(B(x_2,-x_2+4a),C(x_1,-x_1+4a)\)
兩根和\(\displaystyle \frac{-1}{a}\) 兩根積\(\displaystyle \frac{-4a+2}{a}\)
而\(\displaystyle \frac{72}{5}=\frac{1}{2}|\left |\begin{array}{cccc}
0 &x_2 & x_1&0 \\
2 &-x_2+4a &-x_1+4a& 2 \\
\end{array}\right||\)
\(=|2(x_1-x_2)-4a(x_1-x_2)|=|(2-4a)(\sqrt{\frac{1+16a^2-8a}{a^2}})|\)
因為\(\displaystyle a>\frac{1}{2}\)
所以
\(\displaystyle \frac{144}{5}=|(2-4a)(\frac{4a-1}{a})|=(4a-2)(\frac{4a-1}{a})\)
剩下就是解\(a\)
