MathDB

Let

q

be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
[*]In the first line, Gugu writes down every number of the form

a-b

, where

a

and

b

are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form

qab

, where

a

and

b

are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form

a^2+b^2-c^2-d^2

, where

a, b, c, d

are four (not necessarily distinct) numbers from the first line.
Determine all values of

q

such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

Problem Statement