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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 which lie inside a sphere of radius $$10$$ centered at the origin is


Solution

Adding first two equations, we get $$y=0$$
and substituting $$y=0$$ in third equation, we get, $$z=3x$$
So any point which satisfies given system can be taken as, $$(a,0,3a)$$
Now for this point to lie inside inside a sphere of radius $$10$$ centered at origin. 
$$\Rightarrow a^2+0^2+(3a)^2 <10^2$$
$$\Rightarrow a^2<10$$
So, possible integral values of $$a$$ are $$-3,-2,-1,0,1,2,3$$
Hence, number of such points is $$7$$.

Maths

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