x2+y2-2(m+3)+2(1-4m2)y+16m4+9=0一表示一个圆 （1）求该圆半径r的取值范围 （2）求圆心的

(1)x2+y2-2(m+3)x+2(1-4m2)y+16m4+9=0x2-2(m+3)x+(m+3)2+y2+2(1-4m2)y+(1+4m2)2=(m+3)2+(1+4m2)2-16m4-9(x-(m+3)2+(y+(1-4m2)2=m2+6m+9+1+8m2+16m4-16m4-9(x-(m+3)2+(y+(1-4m2)2=9m2+6m+19m2+6m+1>0(3m+1)2>0m≠0m≠-1/3所以m∈R且m≠-1/3(2)圆心坐标为x=m+3m=x-3 1m≠-1/3x-3≠-1/3x≠8/3y=1-4m2 21代入2式得y=1-4(x-3)2=1-4(x2-6x+9)=1-4x2+24x-36=-4x2+24x-37(x∈R,且x≠8/3)所以圆心的轨迹方程是y=-4x2+24x-37(x∈R,且x≠8/3)