A line that's normal to a circle must pass through the center, and any such line works. The center of this circle is (1,2), and an arbitrary line through that point can be constructed with
a(x - 1) + b(y - 2) = 0 . . . . where a and b are not both zero
That's obviously an linear equation in x and y. Further, it's obviously true when x=1 and y=2 so the line includes the point (1,2).
You could also use the point-slope formula, but that wouldn't include the vertical line through (1,2). The above simplifies in to standard form:
ax + by = a + 2b