Prove that all the circles have their centre on a fixed line and passing through a fixed point also pass through a fixed point . through another fixed point

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Dear STudent ,
Not necessarily true - if the fixed point is also on the fixed line.

All circles which have their centres on the fixed line are symmetrical about the fixed line. If two circles intersect (ie pass through a fixed point) above the fixed line, they will intersect also below the fixed line. The only case where this does not happen is where the fixed point is on the line, then the circles touch at the fixed point.
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