(1)
PQ的方程:y = kx + √2
代入椭圆方程,整理得:(2k² + 1)x² + 4√2kx + 2 = 0 (i)
∆ = 32k² - 4*2(2k² + 1) = 16k² - 8 = 0
k² = 1/2
画个草图可知,k > 1/2或k < -1/2
(2)
A(√2,0),B(0,1)
向量AB = (-√2,1)
设P(x₁,y₁),Q(x₂,y₂)
由(i):x₁ + x₂ = -4√2k/(2k² + 1) (ii)
向量OP + 向量OQ = (x₁,y₁) + (x₂,y₂) = (x₁ + x₂,y₁ + y₂) = (x₁ + x₂,k(x₁ + x₂ - 2√2))
若向量OP+OQ与向量AB共线:1/(-√2) = [k(x₁ + x₂ - 2√2)]/(x₁ + x₂)
代入(ii)整理得:2k² + 2k + √2 + 1 = 0
∆ = 4 - 8(√2 + 1) < 0
k不存在