Subjective Type

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 $$7$$.

SIMILAR QUESTIONS

3D Geometry

If $$A (-1, 4, -3)$$ is one end of a diameter $$AB$$ of the sphere $$x^2 + y^2 + z^2 - 3x - 2y + 2z - 15 = 0$$, then find the co-ordinate of $$B$$.

3D Geometry

The equation of the sphere inscribed in a tetrahedron, whose faces are $$x=0,y=0,z=0$$ and $$x+2y+2z=1$$ is

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