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

JEE Advanced · 2009 · 12 Apr 2009 · Paper 2 · Matrices

Let

(x, y, z)

be points with integer coordinates satisfying the system of homogeneous equations:

3 x - y - z - 3 x + z - 3 x + 2 y + z = 0 = 0 = 0.

Then the number of such points for which

x^{2} + y^{2} + z^{2} \leq 100

Answer: 7

From the second equation:

z = 3 x

.

Substituting into the first equation:

3 x - y - 3 x = 0

, so

y = 0

.

The third equation is automatically satisfied.

Thus the solutions are

(x, 0, 3 x)

for any integer

x

.

The constraint

x^{2} + y^{2} + z^{2} \leq 100

becomes:

x^{2} + 0 + 9 x^{2} = 10 x^{2} \leq 100 \Rightarrow x^{2} \leq 10

Thus

x \in {- 3, - 2, - 1, 0, 1, 2, 3}

, giving 7 points.

JEE Advanced · 2009 · 12 Apr 2009 · Paper 2 · Integer answer