(1)∵F(,0), ∴圆心Q在线段OF的垂直平分线x=上 又∵准线方程为:x=-, ∴-(-)=,得p=2, ∴抛物线C:y2=4x; (2)设A(x1,y1),B(x2,y2),过F倾斜角为60°的直线L:y=(x-1). 由得:y2-y-4=0, ∴y1+y2=,y1y2=-4, ∴S△=×|OF|×|y2-y1|=×1×=•=; (3)设直线DE:,可得y2-4my-4t=0,则△=16m2+16t>0(*) 设D(x1,y1),E(x2,y2),则y1+y2=4m,y1y2=-4t, ∵0=•=(x1-4,y1-4)•(x2-4,y2-4)=x1x2-4(x1+x2)+16+y1y2-4(y1+y2)+16 =•-4(+)+16+y1y2-4(y1+y2)+16=-(y1+y2)2+3y1y2-4(y1+y2)+32 =t2-16m2-12t+32-16m, 即t2-12t+32=16m2+16m得:(t-6)2=4(2m+1)2, ∴t-6=±2(2m+1)即:t=4m+8或t=-4m+4 代入(*)式检验均满足△>0, ∴直线DE的方程为:x=my+4m+8=m(y+4)+8或:x=m(y-4)+4, ∴直线过定点(8,-4).(定点(4,4)不满足题意,故舍去) |