Let $(x, y, z)$ be points with integer coordinates satisfying the system of homogeneous equations: $

Question

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

Professor PC Thomas Classes · Answer

(7): $\left.\begin{array}{r}3 x+y-z=0 \ -3 x+z=0\end{array}ight\}$ gives $y=0$\ and $z=3 x$\ we have $x^{2}+y^{2}+z^{2} \leq 100$\ $\Rightarrow \quad x^{2}+9 x^{2} \leq 100$\ $\Rightarrow \quad 10 x^{2} \leq 100 \Rightarrow x^{2} \leq 10$\ $\Rightarrow|x| \leq 3 \Rightarrow-3 \leq x \leq 3$\ Then these are $3-(-3)+1=7$ integral points.\