設(shè)圓心坐標(biāo)為C(a,b),b = a + 1,C(a,a +1)
圓C經(jīng)過(guò)點(diǎn)M（1,3）:(1 -a)^2 + (3 - a -1)^2 = r^2
2a^2 -6a +5 = r^2 (1)
1.圓C與直線L:x-2y-3=0相切,C與直線L距離(d)等于半徑.
d^2 = |a - 2(a+1) -3|^2/(1 + 2^2) = (a+5)^2/5
(a+5)^2/5 = 2a^2 -6a +5
a(9a-40) = 0
a = 0,a = 40/9
A:a = 0:
r^2 = 5
圓心C(0,1)
圓C的方程:x^2 + (y-1)^2 = 5
B:a = 40/9
r^2 = 1445/81
圓心C(40/9,49/9)
圓C的方程:(x-40/9)^2 + (y-49/9)^2 = 1445/81
2.若原點(diǎn)O始終在圓C內(nèi),OC < CM,OC^2 < CM^2
OC^2 = (a - 0)^2 + (a+1 -0)^2 = a^2 + (a+1)^2
CM^2 = (a - 1)^2 + (a + 1 -3)^2 = (a-1)^2 + (a-2)^2
a^2 + (a+1)^2 < (a-1)^2 + (a-2)^2
8a < 4
a < 1/2
r^2 = 2a^2 -6a +5 = 2(a - 3/2) +1/2
r^2是以(3/2.1/2)為頂點(diǎn),開(kāi)口向上的拋物線.a< 1/2時(shí),拋物線為左半部分.
圓C的面積S:πr^2 > π(1/2)^2 = π/4
π/4 < S < +∞