設(shè)圓C的方程為
(x-a)^2 + (y-b)^2 = r^2,
則由圓C過原點(diǎn),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)在圓上,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.
直線x-y+2=0與圓C相切于切點(diǎn)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.
圓C的方程為,
(x-7)^2 + (y+1)^2 = 50.
設(shè)直線L的方程為
y=t-x.t為常數(shù).
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個不同的實(shí)根.
0 t^2 - 12t - 64 = (t-16)(t+4),
-4