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 x2+y2+z2 ≤ 100 is

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 x2+y2+z2 ≤ 100 is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

3 x-y-z=0 -3 x+z=0 ⇒ y=0 and z=3 x ⇒ x2+y2+z2=x2+z2 =x2+9 x2=10 x2 ≤ 100 ⇒ x2 ≤ 10 ⇒ x=0, ± 1, ± 2, ± 3 There are such seven points.

Q. 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

$3 x - y - z = 0$
$- 3 x + z = 0$
$\Rightarrow y = 0$
and $z = 3 x$
$\Rightarrow x^{2} + y^{2} + z^{2} = x^{2} + z^{2}$
$= x^{2} + 9 x^{2} = 10 x^{2} \leq 100$
$\Rightarrow x^{2} \leq 10$
$\Rightarrow x = 0, \pm 1, \pm 2, \pm 3$
There are such seven points.

Q. 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 for which x2+y2+z2≤100 is

3x−y−z=0 −3x+z=0 ⇒y=0 and z=3x ⇒x2+y2+z2=x2+z2 =x2+9x2=10x2≤100 ⇒x2≤10 ⇒x=0,±1,±2,±3 There are such seven points.

Q. 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

$3 x - y - z = 0$
$- 3 x + z = 0$
$\Rightarrow y = 0$
and $z = 3 x$
$\Rightarrow x^{2} + y^{2} + z^{2} = x^{2} + z^{2}$
$= x^{2} + 9 x^{2} = 10 x^{2} \leq 100$
$\Rightarrow x^{2} \leq 10$
$\Rightarrow x = 0, \pm 1, \pm 2, \pm 3$
There are such seven points.