拋物線C:y^2=4x①的焦點為F(1,0),
設(shè)過點K(-1,0)的直線L：x=my-1,
代入①,整理得
y^2-4my+4=0,
設(shè)L與C 的交點A(x1,y1),B(x2,y2),則
y1+y2=4m,y1y2=4,
點A關(guān)于X軸的對稱點D為(x1,-y1).
1.BD的斜率k1=（y2+y1)/(x2-x1)=4m/[m(y2-y1)]=4/(y2-y1),
BF的斜率k2=y2/(x2-1).
k1=k2<==>4(x2-1)=y2(y2-y1),<==>4x2=y2^2,
上式成立,∴k1=k2,∴點F在直線BD上.
2.向量FA*FB=(x1-1,y1)*(x2-1,y2)=(x1-1)(x2-1)+y1y2=(my1-2)(my2-2)+y1y2
=(m^2+1)y1y2-2m(y1+y2)+4=4(m^2+1)-8m^2+4=8-4m^2=8/9,
∴m^2=16/9,m=土4/3.
取y2-y1=√(16m^2-16)=4√（m^2-1)=(4/3)√7,
∴k1=3/√7,BD:y=(3/√7）（x-1).
易知圓心M在x軸上,設(shè)為(a,0),M到x=(4/3)y-1和到BD的距離相等,即
|a+1|/(5/3)=|（3/√7）（a-1)|/(4/√7）,
∴4|a+1|=5|a-1|,-1解得a=1/9.
∴半徑r=2/3,
∴⊿BDK的內(nèi)切圓M的方程為(x-1/9)^2+y^2=4/9.