Question

(1/2)已知方程x^2 +y^2-2(m+3)x+2(1-4m^2)y+16m^4+9=0表示一个圆(1)求实数m的取值范围(2)求该圆的半径

Answer 1

X^2+Y^2-2(m+3)X+2(1-4m^2)Y+16m^4+9=0
(X-(m+3))^2+(Y+1-4m^2)^2=(m+3)^2+(1-4m^2)^2-9-16m^4
等式左边可以化为
左=(m+3)^2+(1-4m^2)^2-9-16m^4
=m^2+6m+9+1-8m^2+16m^4-9-16m^4
=-7m^2+6m+1

1.
左=-(7m+1)(m-1)>0
=> -1/7<m<1

2.
圆的标准方程为(x-a)^2+(y-b)^2=r^2
那么等式右边配方后左边部分应该是半径的平方
要求左边一元二次部分的取值范围要配方，如下
R^2=左=-7(m-3/7)^2+16/7<=16/7
=> 0<R<=4*7^0.5/7 (7^0.5表示根号7)

3.
圆心为(m+3,4m^2-1)
y=4m^2-1
=4*((m+3)^2-6m-9)-1
=4*(m+3)^2-24(m+3)+35
=4x^2-24x+35