Question

Let $S$ 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 $15 \leq x^{2}+y^{2}+z^{2} \leq 150$ Then, the number of elements in the set $S$ is equal to ____.

Answer 1

$\Delta=\begin{vmatrix}1&-2&5\\ -2&4&1\\ -7&14&9\end{vmatrix}=0 $
Let $x= k $
$ \Rightarrow $ Put in (1) & (2)
$ k -2 y +5 z =0$
$ -2 k +4 y + 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
$15 \leq k ^{2}+\left(\frac{ k }{2}\right)^{2}+0 \leq 150$
$12 \leq k ^{2} \leq 120$
$ \Rightarrow \quad k =\pm 4,\pm 6,\pm 8,\pm 10$
$ \Rightarrow $ Number of element in $S =8 $