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

6

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7

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49

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none of these

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Solution

The correct option is B $7$
$3 x - y - z = 0 . . . (1)$
$- 3 x + z = 0 . . . (2)$
$- 3 x + 2 y + z = 0 . . . (3)$
adding (1) and (2) gives $y = 0$
but from (2) $\Rightarrow 3 x = z$
$∴$ solution for the given set of equations is $(x, y, z) = (t, 0, 3 t)$ where $t$ is an integer.
For, $x^{2} + y^{2} + z^{2} \leq 100$
$\Rightarrow 10 t^{2} \leq 100$
$\Rightarrow t^{2} \leq 10$
$∴$ possible values of $t$ are $- 3, - 2, - 1, 0, 1, 2, 3$
i.e $7$ possible solutions.
Hence, option B.

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