Coordinate geometry and perpendicular distance

1) Find the equations of the lines which pass through the point (1,2) and have perpendicular distance 1 from the origin.

The equation of a line passing through (1,2) is y = (2-p) + px for any constant p. I got $\displaystyle |\frac{p-2}{\sqrt(1+p^2)}|=1 $ and therefore $\displaystyle p=\frac{3}{4}$ and the equation of the line is $\displaystyle y=\frac{5}{4} + \frac{3}{4}x$. I just need to check: is this the only possible equation that satisfies the above conditions?