How many points $(x, y)$ with integral co-ordinates are there whose distance from $(1, 2)$ is two units?

$PQ = 2\Rightarrow PQ^{2} = 4$

$(x - 1)^{2} + (y - 2)^{2} = 4$

possibilities for integral coordinates

$\Rightarrow (X - 1)^{2} + (y -2)^{2} = 4 + 0$ or

$= 0 + 4$

Check $(1) (x - 1)^{2} = 4\ (y - 2)^{2} = 0$

$\Rightarrow x - 1 = 2, -2\ ,y - 2 = 0$

$x = 3, -1\ ,y = 2$

$(3, 2)$ and $(-1, 2)$

check $(2) (x - 1)^{2} = 0\ ,(y - 2)^{2} = 4$

$x = 1\ ,y - 2 = 2, -2$

$x = 1\ ,y = 4, y = 0$

$(1, 4)$ and $(1, 0)$

Total $4$ integral coordinates are possible.