Count all the points

Algebra Level 3

Let $(x,y,z)$ be points with integer co-ordinates satisfying the system of homogeneous equations $\begin{cases} 3x-y-z=0 \\ -3x+z=0 \\ -3x+2y+z=0 \end{cases}$ Find the number of such points for which $x^2+y^2+z^2 \leq 100$

You can try more Quadratic equation problems here .

The answer is 7.

2 solutions

Sandeep Bhardwaj
Dec 26, 2014

Given,

$3x-y-z=0$ -------------------(i)

$-3x+2y+z=0$ -------------------(ii)

$-3x+z=0$ ---------------------(iii)

On adding (i) and (ii), we get $y=0$ .

So, $z=3x$ .

Now, $x^2+y^2+z^2 \leq 100$

$\implies x^2+0^2+(3x)^2 \leq 100$

$\implies 10x^2 \leq 100$

$\implies x^2 \leq 10$

$\implies x=-3,-2,-1,0,1,2,3$ .

So, number of such points possible is $7$ .

Note: I apologize for earlier mistake in the question. I'll be extremely careful in the future. Now the question is absolutely right.

enjoy!

Thanks. Those who previously answered 1 (correct answer to the problem as previously phrased) have been marked correct.

Calvin Lin Staff - 6 years, 5 months ago

exactly did the same and i agree with pranjal jain that the problem is quite overrated but your set of jee maths :quadratic equations i awesome

Parv Maurya - 6 years, 3 months ago

Another way, albeit same as yours, is to use matrix theory to solve the system of equations.

Prasun Biswas - 6 years, 3 months ago

Did the same way!!😀😀

Anurag Pandey - 4 years, 10 months ago

Mayank Singh
Feb 12, 2015

Use an 2 equations to get y=0 and z=3x Now substitute it in the equation of the sphere, to get x^2 <= 10

Count all the points

The answer is 7.

2 solutions

0 pending reports