～ところを— Caught In The Act

• 授業中に居眠りしているところを先生に見られた。

  Jugyou-chuu ni inemuri shite iru tokoro o sensei ni mirareta.

  I was caught by the teacher dozing off during class.

  ところを makes the dozing situation the object of 見られた (was seen).

• 泥棒は逃げるところを警察に捕まった。

  Dorobou wa nigeru tokoro o keisatsu ni tsukamatta.

  The thief was caught by police just as he was fleeing.

  ところを marks the fleeing moment that was interrupted by the arrest.

• お忙しいところをすみません。

  Oisogashii tokoro o sumimasen.

  Sorry to bother you when you're busy.

  ところを is a set expression apologizing for imposing on someone's situation.

• ・The main verb is usually one of seeing, catching, stopping, or helping (見る, 捕まえる, 止める, 助ける).
• ・The set phrase お忙しいところを / お休みのところを is polite and very common.
• ・Register: neutral to polite; the apology form is quite formal.

Commonly confused with

The JLPT rarely asks whether you recognise a pattern — it asks which of several similar patterns fits. These are the ones ～ところを is tested against.

• ～ないうちに — Before It Happens
• ～上(で) — After Doing / In the Process Of
• ～から～にかけて — Spanning From To
• ～際(に) — On The Occasion Of
• 最中に — Right in the Thick of It
• 次第 — The Moment It's Done
• ～て以来 — Ever Since Then
• ～ところに／～ところへ — Right At That Moment