填充七
若兩直線在\( y=ax^2 \)的頂點\(O\)互相垂直,且分別與拋物線交於\(A\)、\(B\)兩點,若\( \Delta OAB \)的最小面積為4,則\( a= \) 。
[解答]
假設坐標\(\displaystyle A(t,at^2)、B(s,as^2) \)
要滿足\(\displaystyle ts+a^2t^2s^2=0 \)
也就是\(\displaystyle a^2ts=-1 \)
可以知道\(\displaystyle t,s \)一正一負
計算三角形OAB面積\(\displaystyle (OAB) \)
\(\displaystyle =\frac{1}{2}|ats^2-at^2s| \)
\(\displaystyle =\frac{1}{2}|ats||s-t| \)
\(\displaystyle =\frac{1}{2|a|}|t-s| \)
不妨假設\(\displaystyle t>0>s \)
\(\displaystyle t-s=t+(-s)\ge 2\sqrt{\frac{1}{a^2}}=\frac{1}{|a|} \)
所以三角形OAB面積的最小值就是
\(\displaystyle \frac{1}{a^2} \)
所以
\(\displaystyle \frac{1}{a^2}=4 \)
\(\displaystyle a=\pm\frac{1}{2} \)