已知圆x^2+y^2+x-6y+m=0与直线x+2y-3=0交与P、Q两点,O为坐标原点,问是否存在实数m使得OP⊥OQ

1个回答

  • 将圆方程化简为标准式有:

    [x+(1/2)]^2+(y-3)^2=(37-4m)/4……………………………(1)

    所以,圆心坐标为(-1/2,3)

    联立直线与圆方程得到:

    x^2+x+y^2-6y+m=0

    x+2y-3=0

    ===> (2y-3)^2-(2y-3)+y^2-6y+m=0

    ===> 4y^2-12y+9-2y+3+y^2-6y+m=0

    ===> 5y^2-20y+(m+12)=0

    ===> y1+y2=4,y1y2=(m+12)/5

    ===> x1x2=(-2y1+3)(-2y2+3)=4y1y2-6(y1+y2)+9=4(m+12)/5-15

    已知OP⊥OQ

    则,Kop*Koq=-1

    即:(y1/x1)*(y2/x2)=-1

    ===> y1y2+x1x2=0

    ===> (m+12)/5+4(m+12)/5-15=0

    ===> m+12-15=0

    ===> m=3