Answer : False
The equation of given circle is $x^2+y^2-2x+6y+1=0$
The given point $P(1,2)$
The coordinates of the centre of the circle is $C(1,-3)$
$\therefore CP=\sqrt{(1-1)^2+(2+3)^2}$
$\Rightarrow 5$
Since $5>0$,the point lies outside the circle.
Hence the given statement is False.