已知圆c经过坐标原点,且与直线x-y+2=0相切,切点为A(2.4).若斜率为-1的直线L与圆c相交与两个不同的点M.N

设圆C的方程为(x-a)^2 + (y-b)^2 = r^2,则由圆C过原点,有a^2 + b^2 = r^2,(x-a)^2 + (y-b)^2 = a^2 + b^2,x^2 -2ax + y^2 -2by = 0.A(2,4)在圆上,有0 = 2^2 - 2a*2 + 4^2 - 2b*4 = 4 - 4a + 16 - 8b = 4[5-a-2b].5=a+2b,a=5-2b.x^2 -2(5-2b)x +y^2-2by=0.直线x-y+2=0与圆C相切于切点A(2.4),有0 = 2x -2(5-2b) + 2y*y' - 2b = 2[x+y*y'-5+b].y'(2)=1.y(2)=4.0 = 2[2 + 4*1 -5 + b]=2(1+b),b=-1,a=5-2b=5+2=7,r^2=a^2+b^2=7^2 + 1=50.圆C的方程为,(x-7)^2 + (y+1)^2 = 50.设直线L的方程为y=t-x.t为常数.50 = (x-7)^2 + (t-x+1)^2 = (x-1)^2 -12(x-1) + 36 + t^2 -2t(x-1) + (x-1)^2 = 2(x-1)^2 - 2(x-1)[6+t] + t^2 + 36,方程0 = (x-1)^2 - (x-1)(6+t) + t^2/2 - 7有2个不同的实根.0 t^2 - 12t - 64 = (t-16)(t+4),-4