Question

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

• A. One
• B. Two
• C. Three
• D. Four

Answer 1

The correct option is D Four

$PQ=2\Rightarrow P{Q}^2=4$
$(x-1{)}^2+(y-2{)}^2=4$
possibilities for integral coordinates
$\Rightarrow (X-1{)}^2+(y-2{)}^2=4+0$ or
$=0+4$
Check $\left(1\right)(x-1{)}^2=4(y-2{)}^2=0$
$\Rightarrow x-1=2,-2,y-2=0$
$x=3,-1,y=2$
$(3,2)$ and $(-1,2)$
check $\left(2\right)(x-1{)}^2=0,(y-2{)}^2=4$
$x=1,y-2=2,-2$
$x=1,y=4,y=0$
$(1,4)$ and $(1,0)$
Total $4$ integral coordinates are possible.