Question

一个圆过定点P(2,-1)和直线x-y=1相切,并且圆心在y=-2x上,求它的方程.

Answer 1

设圆心坐标是O(m,-2m)

圆心O到点P的距离应等于圆心到直线x-y-1=0的距离

(m-2)²+(-2m+1)²=(m+2m-1)²/(1²+(-1)²)

m²-4m+4+4m²-4m+1=(9m²-6m+1)/2

5m²-8m+5=(9m²-6m+1)/2

m²-10m+9=0

(m-1)(m-9)=0

解得m=1或m=9

圆心坐标是(1,-2)或(9,-18)

圆的半径是根号2或13根号2

圆的方程是(x-1)²+(y+2)²=2或(x-9)²+(y+18)²=338