Why is the line "At home, my lord" so important, and why might it make Hamlet angry?

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The line itself isn't immediately important or significant. It's not like it's a key theme of the play, really, but it's a significant moment in the scene.

Hamlet and Ophelia have been having a conversation about something else entirely, when suddenly Hamlet changes the topic:

HAMLET: Where's your father?

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The line itself isn't immediately important or significant. It's not like it's a key theme of the play, really, but it's a significant moment in the scene.

Hamlet and Ophelia have been having a conversation about something else entirely, when suddenly Hamlet changes the topic:

HAMLET:
Where's your father?

OPHELIA:
At home, my lord.

Claudius and Polonius are hiding in the same room. Has Hamlet suddenly realised that Polonius is watching him? Has Ophelia just somehow given the game away? That varies from production to production. But Ophelia's answer makes Hamlet change topic again, and suddenly start verbaly attacking her. Why? Because, perhaps, it is a lie - a direct lie - and Ophelia knows it. Because, perhaps, the way she says it tells Hamlet that Polonius is watching.

But it opens a little window into the deception is going on in the scene - and perhaps Hamlet does recognise that.  

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If one is reading the play as though Hamlet is sane and merely manipulating his surroundings, then "At home, my lord" becomes Hamlet's confirmation that Ophelia is supporting her conniving father and not Hamlet, because he knows she is lying to him, and that Polonius is still in the castle, and possibly even knows that he is listening.

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