当a为何值时,直线x+y-2a+1=0与圆x^2+y^2-2x+4y+1=0相切?相离?相交?

Question

当a为何值时,直线x+y-2a+1=0与圆x^2+y^2-2x+4y+1=0相切?相离?相交?

Answer 1

圆方程可变为 (x - 1)² + (y + 2)² = 4,圆心C(1,-2),半径r = 2
C与直线的距离d = |1 - 2 - 2a + 1|/√2 = √2|a|
(1) 相切
d = r,√2|a| = 2,a = ±√2
(2) 相离
d > r
√2|a| > 2,a > √2 或 a < - √2
(3)相交
d < r
√2|a| < 2,-√2 < a < √2