动圆C1:x2+y2=t2,1

(1)
y² = t²- x²
代入x²/9+y²=1
x² = 9(t² - 1)/8
y² = (9 - t²)/8
矩形ABCD的边长分别为2|x|,2|y|
面积S,S² = (2|x|*2|y|)² = 16x²y² = (9/4)(t² - 1)(9 - t²)
= -(9/4)(t² - 5)² + 36
t² = 5,S最大,为6
(2)
A1(-3,0),A2(3,0)
不妨取p = √[9(t² - 1)/8],q = √[(9 - t²)/8]
A(p,q),B(p,-q)
AA1的方程:(y - 0)/(q - 0) = (x + 3)/(p + 3)
BA2的方程:(y - 0)/(-q - 0) = (x - 3)/(p - 3)
联立,x = 9/p,p = 9/x (i)
y = 3q/p,q = py/3 = (9/x)y/3 = 3y/x (ii)
p² = 9(t² - 1)/8 = 81/x²
t² = 1 + 72/x² (iii)
q² = (9 - t²)/8
代入(ii)(iii)并整理:x²/9 - y² = 1

由对称性可知，A1A和A2B焦点M一定在y轴上。