圆C1:x^2+y^2=r^2截直线x+y-√3/2=0所得的弦长为√3,
弦心距d=√6/4,满足2√(r^2-d^2)=√3,
∴r^2-3/8=3/4,r^2=9/8,
C1:x^2+y^2=9/8,
抛物线C2:x^2=2py(p>0)的焦点(0,p/2)在圆C1上,
(p/2)^2=9/8,p=3√2/2.
过点A（0,2）的直线l:y=kx+2①与抛物线C2:x^2=3√2y②交于B,C两点,
把①代入②,x^2-3√2kx-6√2=0,
△=18k^2+24√2,
设B(x1,y1),C(x2,y2),则x1+x2=3√2k,x1x2=-6√2,
又过B,C两点分别作抛物线C2的切线:x1x=(3√2/2)(y1+y)③,x2x=(3√2/2)(y2+y),④
(1)两条切线的斜率之积=x1x2/[3√2/2]^2=-6√2/（9/2）=-4√2/3,为定值.
(2)|BC|=√[△(1+k^2)],两条切线交于D点：
③*x2-④*x1,0=(3√2/2)(x2y1+x2y-x1y2-x1y),
∴(x2-x1)y=x1y2-x2y1=x1(kx2+2)-x2(kx1+2)=-2(x2-x1),
∴y=-2,
代入③,x1x=(3√2/2)(y1-2)=(3√2/2)kx1,x=(3√2/2)k,
∴D到l的距离h=|(3√2/2)k^2+4|/√(k^2+1),
∴S△BCD=(1/2)|BC|h=[(3√2/4)k^2+2]√(18k^2+24√2),是k^2的增函数,
∴k=0时它取最小值2√(24√2).