Take the voltage out of a voltage divider with no distortion
This question is usually addressed with a buffer amplifier.  The goal of the buffer amplifier circuit is to have a high impedance input and low impedance output.

A very common way of doing this is to wire an OpAmp in a voltage follower configuration.

 (source of the picture)

Simply put, if you connect the voltage divider to Vin and connect the 10Ω between Vout and ground, then the voltage divider will not "see" the additional 10Ω in parallel with 1kΩ.

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The potential problem with 10Ω (low impedance load) is that it will draw current (I=Vout/Rload) from the OpAmp output, and the OpAmp has to be able be able to supply it.


Let's assume a scenario in which the voltage after the divider is 0.5V.  Then the OpAmp will need to supply 50mA of output current.  This falls into the the specs of a typical vanilla OpAmp.
Assume another scenario in which the voltage after the divider is 6V.  Then the OpAmp will need to supply 600mA of output current.  This is outside the spec of a typical OpAmp (although this still can be achiehved with an additional high current stage).


BTW.  I'll hazard a guess that you'll have 0.312V max after your 1:1001 voltage divider, if you apply 220 VAC to the voltage divider.  220 VAC is an RMS value.