The locus of midpoints of the chords of the circle
\[
x^2+y^2-2x-2y+1=0
\]
which are parallel to the line
\[
x+y+2=0
\]
is
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The line joining the centre of a circle to the midpoint of a chord is always perpendicular to the chord.
Hence, the locus of the midpoints of a family of parallel chords is a diameter perpendicular to those chords.