In Othello, Act 1, Scene 1, why does Iago leave Roderigo alone at Brabantio's house?
Iago leaves Roderigo alone at Brabantio's house because he says that it is not fitting for a man of his inferior rank to be caught sullying the name of Othello. Iago says,
"It seems not meet, nor wholesome to my place, to be produced - as, if I stay, I shall - against the Moor...Though I do hate him as I do hell-pains, yet for necessity of present life, I must show out a flag and sign of love, which is indeed but sign" (lines 158-160, 167-170).
In simple paraphrase, Iago is saying that it would not be right for him to be found speaking again Othello, the Moor, and that even though he hates him, he must act as if he loves him, to preserve his current position in life. In reality, Iago is plotting against Othello, and, in setting his plan in motion, enlists the aid of Roderigo, who is a rejected suitor of the beautiful Desdemona. Iago asks Roderigo to go to Brabantio, Desdemona's father, to tell him he has been robbed. According to Iago, not only have Brabantio's possessions been taken, but his daughter is gone as well, having eloped with Othello. Roderigo describes a hideous scene of the violation of Desdemona to her father, inflaming his rage against Othello, and Iago, his purposes accomplished, discreetly removes himself from the scene to warn Othello of Branbantio's ire so that his own complicity in the plot will not be uncovered, leaving Roderigo to deal with an angry Brabantio (Act I, Scene 1).
As you will observe after you get further into the play, Iago is two-faced and must maintain a certain appearance (one of loyalty to Othello) in order to accomplish his goal of supplanting Othello and Cassio. He sends Roderigo alone so that Brabantio will not know that Iago has any part in bringing the news of Desdemona's elopement. Iago is Othello's right-hand man, and it would seem disloyal and strange if he were the one to stir up Othello's new father-in-law against him.
You will continue to see Iago use Roderigo in such a way so that Iago can maintain his reputation of loyalty and submission to Othello.