Let $\mathrm{S}$ be the set of all integer solutions, $(x, y, z)$, of the system of equations
$x-2 y+5 z=0$
$-2 x+4 y+z=0$
$-7 x+14 y+9 z=0$
such that $15 \leq \mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2} \leq 150$. Then, the number of elements in the set $S$ is equal to__________
$\Delta=\left|\begin{array}{ccc}1 & -2 & 5 \\ -2 & 4 & 1 \\ -7 & 14 & 9\end{array}\right|=0$
Let $\quad x=k$
$\Rightarrow \quad$ Put in (1) \& (2)
$k-2 y+5 z=0$
$-2 k+4 y+z=0$'
$z=0, y=\frac{k}{2}$
$\therefore \quad \mathrm{x}, \mathrm{y}, \mathrm{z}$ are integer
$\Rightarrow \quad \mathrm{k}$ is even integer'
Now $\mathrm{x}=\mathrm{k}, \mathrm{y}=\frac{\mathrm{k}}{2}, \mathrm{z}=0$ put in condition
$15 \leq \mathrm{k}^{2}+\left(\frac{\mathrm{k}}{2}\right)^{2}+0 \leq 150$
$12 \leq \mathrm{k}^{2} \leq 120$
$\Rightarrow \quad \mathrm{k}=\pm 4, \pm 6, \pm 8, \pm 10$
$\Rightarrow$ Number of element in $\mathrm{S}=8$