ところを

ところを takes a moment or a state – someone busy, resting, about to leave – and makes it the object of what happens next. Its commonest use is in polite set phrases that apologise for, or thank someone for, an imposition: おいそがしい忙しいところを, "when you are busy".

These phrases belong to business and formal speech. In speech the を often drops: おいそがしい忙しいところすみません.

Caught in the act

After a ている verb or a ようとした verb, ところを marks the moment someone is seen, caught, stopped, woken or helped. The next verb acts on that moment, and it is often passive.

ところを or ところに

With ところに, something arrives into the moment. With ところを, the moment is what the next verb acts on.

• × 寝ているところに母に見られた → 寝ているところを母に見られた

"Where you would normally"

With ほんらい本来なら or ふつう普通なら, ところを says what would normally have happened, and the second half says what happened instead.