过点M(4,0)的直线l与抛物线y²=4x的两个交点为A,B 证明:以A,B为直径...答:设直线l:x=my+4,A(x1,y1),B(x2,y2),把直线方程代入抛物线方程,消去x整理得:y²-4my-16=0,∴y1+y2=4m, y1·y2=-16,∴向量OA·向量OB=x1·x2+y1·y2 =(my1+4)(my2+4)+y1·y2 =(m²+1)y1·y2+4m(y1+y2)+16 =-16(m²+1)+4m·(-4m)+16=0...
过抛物线焦点y^2=4x的焦点F作直线l与抛物线交于A,B两点。求证:△AOB...答:抛物线y^2=4x的焦点为F(1,0),设过F的直线l:x=my+1,① 代入y^2=4x,整理得 y^2-4my-4=0,设A(x1,y1),B(x2,y2),则 y1+y2=4m,y1y2=-4.由①,x1x2=(my1+1)(my2+1)=m^2y1y2+m(y1+y2)+1 =-4m^2+4m^2+1=1,∴向量OA*OB=x1x2+y1y2=-3,向量AO*AB=(...