Question

The number of the points in the cartesian plane with integral coordinates satisfying the inequalities $|x|\le k$ , $|y|\le k$ , $|x-y|\le k$ is $($where $k\in N)$

• A. $(k+1{)}^3-k^3$
• B. $(k+2{)}^3-(k+1{)}^3$
• C. $(k^2+1)$
• D. None of these

Answer 1

The correct option is A $(k+1{)}^3-k^3$

$|x|\le k$
$\Rightarrow -k\le x\le k\cdots \left(i\right)$
and $|y|\le k$
$\Rightarrow -k\le y\le k\cdots \left(ii\right)$
and $|x-y|\le k$
$\Rightarrow |y-x|\le k$
$\Rightarrow -k\le y-x\le k$
$\Rightarrow x-k\le y\le x+k\cdots \left(iii\right)$


Number of the points having integral coordinates
$=(2k+1{)}^2-2[k+(k-1)+...+2+1]$
$=3k^2+3k+1$
$=(k+1{)}^3-k^3$