Question

Let $S$ be the set of all integer solutions, $(x,y,z),$ of the system of equations
$x-2y+5z=0$
$-2x+4y+z=0$
$-7x+14y+9z=0$
such that $15\le x^2+y^2+z^2\le 150$ Then, the number of elements in the set $S$ is equal to ____.

Answer 1

Answer: 8

$\Delta =\left|\begin{array}{ccc}1& -2& 5\\ -2& 4& 1\\ -7& 14& 9\end{array}\right|=0$
Let $x=k$
$\Rightarrow$ Put in (1) & (2)
$k-2y+5z=0$
$-2k+4y+z=0$
$z=0,y=\frac{k}{2}$
$\therefore x,y,z$ are integer
$\Rightarrow k$ is even integer
Now $x=k,y=\frac{k}{2},z=0$ put in condition
$15\le k^2+{\left(\frac{k}{2}\right)}^2+0\le 150$
$12\le k^2\le 120$
$\Rightarrow k=\pm 4,\pm 6,\pm 8,\pm 10$
$\Rightarrow$ Number of element in $S=8$