There was no episode that shows the night they conceive Emma. We only learn the details in retrospect , in season 8 episode 4 "The One with the Videotape". We learn that they conceived Emma about 6 weeks before they show the tape.
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Emma is sent to stay with her grandmother (Rachel's Mum) while Rachel moves to Paris and then is supposed to be brought over to Rachel or Rachel would go and get her. But because she doesn't actually move to Paris in the end, Emma is brought back to Rachel to live with her and Ross.
She gets anorexia and she does not die. I needed to get that out of head because people have been making videos and saying that she does.
But JT does get stabbed and dies and so does Rick.