Question

Let $A=(3,4)$ and $B=(6,\beta )$. If the length of the line segment joining $A$ and $B$ is less than or equal to $4$, then the number of integral value(s) of $\beta$ is

Answer 1

Distance between $A$ and $B$ is,
$\sqrt{(3-6{)}^2+(4-\beta {)}^2}\le 4$
Squaring both sides,
$0\le 9+(\beta -4{)}^2\le 16\Rightarrow -9\le (\beta -4{)}^2\le 7\Rightarrow -\sqrt{7}\le \beta -4\le \sqrt{7}\Rightarrow 4-\sqrt{7}\le \beta \le 4+\sqrt{7}$

Therefore, the integral values of $\beta$ are $2,3,4,5,6$
Hence, the number of integral values of $\beta$ is $5$.