(1)
A(0,-8) ,C(c,0),M(0,m),∠C=90°,B(x,y)
AM = (1/2)(AB+AC)
OM-OA =(1/2)(OB-OA +OC -OA)
(0,m+8) = (1/2)( (x,y) + (c,16))
= (1/2)(x+c,y+16)
=> (1/2)(x+c)=0
c = -x
∠C=90°
=> AC.CB =0
(c,8).(x-c,y) =0
c(x-c) - 8y = 0
-x(2x) -8y =0
x^2 + 4y =0
E的方程:x^2 + 4y =0 (1)
(2)
let Q(x1,y1),N(x2,y2)
QP=PN
OP-OQ = ON-OP
(2-x1,4-y1)= (x2-2,y2-4)
=> 2-x1= x2-2 and 4-y1 = y2-4
=> x1+x2 = 4 and y1+y2 = 8
l:y= mx + c
过点P(2,4）
=> 4 = 2m +c
c = 4-2m
l:y = mx+ 4-2m (2)
sub (2) into (1)
x^2 + 4y =0
x^2 + 4(mx+ 4-2m) =0
x^2 + 4mx + 16-8m =0
x1+x2 = -4m = 4
=> m = -1
l:y = mx+ 4-2m
y = -x + 6