Let (x, y, z) be points with integer coordinates satisfying system of homogeneous equations: 3 x-y-z

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3 and 4 .Determinants and Matrices

hard

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

$3$

$9$

$5$

$7$

(IIT-2009)

Adding first two equations, we get $y = o$

and substituting $y=0$ in third equation, we get, $z=3 x$

So any point which satisfies given system can be taken as, $(a, o, 3 a )$

Now for this point to lie inside inside a sphere of radius $10$ centered at origin.

$\Rightarrow a ^2+ o ^2+(3 a )^2 < 10^2$

$\Rightarrow a ^2 < 10$

So, possible integral values of a are $-3,-2,-1,0,1,2,3$

Hence, number of such points is $7$ .

Standard 12

Mathematics

medium

hard

(JEE MAIN-2020)

normal

medium

hard

(JEE MAIN-2020)