Because b is on the ellipse,

So x2? + 2 y2？ =4

So y2? ≤2，-y2？ ≥-2

|OA|？ = x 1？ + 4

|OB|？ = x2？ + y2？ = 4 - y2？ ≥0, so

|AB|？ = x 1？ + 4 + 4 - y2？ = x 1？ - y2？ + 8≥x 1？ +6≥6

So the shortest line segment length of AB is √6.

I didn't work out 2√2. Oh ╮ (╯ ▽ ╰) ╭ If you see which step is wrong, please tell me and make progress together.