Question

Let $(x,y,z)$ be points with integer coordinates satisfying the system of homogeneous equations:
$3x-y-z=0-3x+z=0-3x+2y+z=0$
Then the number of such points for which
$x^2+y^2+z^2\le 100$ is ___

Answer 1

The given system of equations is -
$3x-y-z=0-3x+z=0-3x+2y+z=0$
Let $x=p$ where $p$ is an integer, then $y=0andz=3p$
$but{x}^2+y^2+z^2\le 100\Rightarrow p^2+9p^2\le 100\Rightarrow p^2\le 10\Rightarrow p=0,\pm 1,\pm 2\pm 3$
i.e. $p$ can take $7$ different values.
$\therefore \text{Number of points}(x,y,z)\text{are}7$.