1)
y=ax²,则y'=2ax
P(2,n)处切线斜率为k=4a,且n=4a
P点处切线与圆相切,则圆心M(3,0)和P连线斜率为-1/4a
则(0-4a)/(3-2)=-1/4a
a=1/4 (a=-1/4舍去)
2)
y=1/4x² P(2,1)
先猜再证,猜m点为抛物线焦点(0,1)
设A(x1,y1) B(x2,y2),AB为y=kx+1
则由抛物线定义|Am|=A到准线距离=y1+1
|Bm|=y2+1
联立得到：y²-(4k²+2)y+1=0
y1+y2=4k²+2 y1y2=1
1/|Am|+1/|Bm|
=1/(y1+1)+1/(y2+1)
=(y1+y2+2)/(y1y2+y1+y2+1)
=(4k²+4)/(4k²+4)
=1为定值
故m坐标为(0,1) M=1满足题意