Question

Let $A=(3,4),B$ is a variable point on the lines $\left|x\right|=6.$ If $AB\le 4,$ then the number of positions of $B$ with integral coordinates is

• A. $5$
• B. $4$
• C. $6$
• D. $10$

Answer 1

The correct option is A $5$

$A(3,4)$

Let the point be $B(\pm 6,y)$ since $\left|x\right|=6$

Case 1: $x=6$

$\sqrt{{(6-3)}^2+{(y-4)}^2}<4\Rightarrow 9+y^2+16-8y<16\Rightarrow y^2-8y+16<0$

When $y^2-8y+16=0\Rightarrow y=4\pm \sqrt{7}$

$\therefore y\in (4-\sqrt{7},4+\sqrt{7})\Rightarrow y\in (1.34,6.64)$

Case 2: $x=-6$
$\therefore B(-6,y)$

$\therefore \sqrt{{(-6-3)}^2+{(y-4)}^2}<4\Rightarrow 81+y^2+16-8y<16$

$\Rightarrow y^2-8y+81<0$

which is not reduce in factors

Therefore points are $(6,2),(6,3),(6,4),(6,5)$ and $(6,6)$